HSF.IF.C.9: Comparing Functions in Different Representations
In plain English: HSF.IF.C.9 is the Common Core functions standard that asks students to compare properties of two functions when each is given in a different way: as an equation, a graph, a table or a verbal description. Students compare features such as intercepts, maximum and minimum values, rates of change and growth. It is usually taught in Algebra I and applied again in Algebra II.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Common Core State Standards for Mathematics · Domain: Interpreting Functions (IF) · Cluster: Analyze functions using different representations Also written as HSF-IF.C.9 or F-IF.9 · Official standard
Students compare properties of two functions when each one is given in a different way: algebraically, graphically, numerically in a table, or by a verbal description. The properties are the ones students already know how to find: intercepts, maximum and minimum values, rates of change, and how the functions grow in the long run. The new skill is to find the same property in two different representations and put the two values side by side.
The lesson opens with the official example, a quadratic given by a graph against one given by an expression, and then moves through tables and verbal descriptions with linear, quadratic and exponential functions. Students justify each comparison with a value from each function, not with a general impression of which one "looks bigger."
Learning Objectives
By the end of this lesson, students will be able to:
Find intercepts, maximum or minimum values and rates of change from an equation, a graph, a table and a verbal description
Compare the same property of two functions that are given in different representations
Decide which of two quadratic functions has the larger maximum or smaller minimum when one is given by a graph and the other by an expression
Compare linear and exponential functions given in different ways, including which one is greater for large inputs
Prior Knowledge Required
Students should already be comfortable with:
Comparing two linear functions given in different ways 8.F.A.2
Interpreting key features of graphs and tables HSF.IF.B.4
Calculating average rate of change from a table or an equation HSF.IF.B.6
Completing the square to find the vertex of a quadratic HSF.IF.C.8
Show two functions side by side, one as an equation and one as a table:
Warm-Up Prompt
"f(x) = 3x - 2. For g, the table gives x = 0, 1, 2 and g(x) = 1, 5, 9. Which function increases faster? Which has the greater value when x = 0? How did you find each answer from each representation?"
Collect answers: f has slope 3 (read from the equation); g increases by 4 each time x increases by 1 (read from the table), so g increases faster. At x = 0, f(0) = -2 and g(0) = 1, so g is greater. Point out that students answered the same two questions with different tools. That is the skill of the lesson: find the same property in each representation, then compare.
Direct Instruction25 minutes
Build a class chart: for each property, where do you find it in each representation?
Intercepts: in an equation, substitute x = 0 or solve f(x) = 0; on a graph, read where it crosses the axes; in a table, look for x = 0 and for f(x) = 0 (or a sign change); in words, look for "starts at" or "reaches zero when".
Maximum or minimum: complete the square or use x = -b/(2a); on a graph, read the highest or lowest point; in a table, use the symmetry of the outputs; in words, look for "greatest", "highest" or "lowest".
Rate of change: the slope or growth factor in an equation; rise over run on a graph; differences or ratios of outputs in a table (check that the x-steps are equal); "per", "each" or "percent" in words.
Long-run behavior: decide whether each function is linear, quadratic or exponential, then compare how they behave for large inputs.
Work the official example first with Diagram 1, then the other three examples. Each time, have students write the property of each function in the same form (for example, "maximum = 6" and "maximum = 7") before comparing.
Official example: graph and expression
Function f is given by the graph in Diagram 1, a parabola through (0, 0), (2, 6) and (4, 0). Function g is g(x) = -2x² + 4x + 5. Which has the larger maximum?
Equation: From the graph, f has maximum 6 at x = 2. Completing the square, g(x) = -2(x - 1)² + 7, so g has maximum 7 at x = 1. g has the larger maximum.
Expression and table: rates of change
Ride service A charges C(d) = 2.50 + 1.25d dollars for d miles. Service B's prices are in a table: 0 miles, $4.00; 2 miles, $6.00; 4 miles, $8.00; 6 miles, $10.00. Compare the two.
Equation: A: $1.25 per mile, $2.50 base. B: (6.00 - 4.00)/2 = $1.00 per mile, $4.00 base. A starts lower but rises faster; 2.50 + 1.25d = 4 + d gives d = 6, where both cost $10.00
Verbal description and table: linear and exponential
Town A has 8,000 residents and gains 300 residents per year. Town B's population is in a table: year 0, 6,000; year 1, 6,600; year 2, 7,260; year 3, 7,986. Compare initial values and growth.
Equation: A starts larger and grows by a constant 300 per year. B grows by a constant factor, 1.1 (10% per year). Year 4: A 9,200, B about 8,785; year 5: A 9,500, B about 9,663, so B passes A during year 5
Expression and verbal description: maximum and zeros
Rocket A has height h(t) = -16t² + 64t + 80 feet. Rocket B is launched from the ground and reaches its greatest height, 196 feet, after 3.5 seconds. Which goes higher, and which lands first?
Equation: A: h(t) = -16(t - 2)² + 144, maximum 144 ft; h(t) = 0 at t = 5. B: maximum 196 ft, and by symmetry it lands at t = 7. B goes higher; A lands first
End with a warning about tables: a table shows only some inputs. The largest value in a table is not always the maximum of the function, so students should use a pattern, such as symmetric outputs or constant differences, to justify what happens between and beyond the rows.
Guided Practice15-20 minutes
Pairs solve three comparisons, and the class discusses each before moving on: (1) Candle A is 30 cm tall and burns 2 cm per hour; candle B has height h(t) = 24 - 1.5t cm. Which burns out first? (A after 15 hours, B after 16 hours.) (2) A table gives x = 0, 1, 2, 3 and y = 1, 3, 9, 27; compare it with y = 10x + 1 at x = 3 and at x = 4. (27 < 31, but 81 > 41.) (3) Parabola P is described as "opens down, zeros at -2 and 6, maximum value 8"; compare it with k(x) = -x² + 2x + 10. (P: axis x = 2, maximum 8; k: axis x = 1, maximum 11.) Listen for students who compare a slope with an initial value, or who read the maximum of P as 6 because 6 is a zero.
Independent Practice15 minutes
Students work alone on four comparisons: (1) f(x) = -3x + 12 and a line described as "crosses the y-axis at 9 and the x-axis at 6": compare slopes (-3 and -1.5) and x-intercepts (4 and 6). (2) A table of a quadratic, x = 0, 1, 2, 3, 4 and y = 8, 3, 0, -1, 0, against m(x) = (x - 1)² - 3: compare minimum values (-1 and -3). (3) A savings account with $500 growing 4% per year against a jar that starts with $600 and gains $15 per year: compare after 1 year ($520 and $615) and after 20 years (about $1,096 and $900). (4) Choose any two functions from the class examples and write two true comparison sentences, one about a feature and one about a rate.
Closure5-10 minutes
Exit ticket: (1) f(x) = -(x + 2)² + 9, and g is a parabola that opens down with vertex (4, 10). Which has the greater maximum? (g: 10 > 9.) (2) A table shows x = 0, 1, 2 and y = 7, 11, 15; compare its rate of change with y = 5x - 3. (4 per unit and 5 per unit.) (3) Which representation made each property easiest to find today, and why?
Differentiation Strategies
For Struggling Students
Provide the class chart as a handout: one row per property, one column per representation, with a worked entry in each column
Have students convert both functions to the same representation first, such as a short table, then compare; later, remove this step
Limit the first comparisons to linear functions, then add quadratics and exponentials
For Advanced Students
Ask students to write a function in a new representation that ties a given function on two properties at once, such as the same maximum and the same y-intercept
Give two functions where the answer depends on the interval, such as a linear and an exponential function, and ask for the exact interval where each is greater
Ask for a table of values that could come from two different functions with different maximums, and explain why a table alone cannot settle the comparison
Assessment Guidance
What to Look For
A complete comparison names the property, gives a value for each function and states which is greater: "f has maximum 6 and g has maximum 7, so g has the larger maximum." Watch for students who compare different properties (the slope of one with the y-intercept of the other), who take the largest number in a table as the maximum without checking the pattern, and who assume equal x-steps in a table that has unequal steps.
02
Classroom Activities
3 Activities
1
Representation Showdown
25 minGroups of 4
Each group gets 16 function cards arranged as 8 pairs, and each card in a pair shows its function in a different representation. For each pair, the group answers the question printed on the pair and justifies it with one value from each card.
The Eight Pairs (teacher key)
y = 7x + 2 vs. a line through (0, 5) and (2, 15): greater slope? (7 > 5, the equation)
A table with x = 0, 1, 2, 3 and y = 12, 9, 6, 3 vs. y = -2x + 10: greater x-intercept? (4 and 5, the equation)
y = x² - 8x + 20 vs. "a parabola that opens up with lowest point (3, 5)": smaller minimum? (4 < 5, the equation)
y = -x² - 2x + 3 vs. a table with x = -1, 0, 1, 2, 3 and y = 0, 3, 4, 3, 0: greater maximum? (4 for both: a tie)
"$50 that doubles every year" vs. y = 80(1.5)x: greater after 3 years? ($400 > $270, the description)
A graph through (0, 6), (1, 3) and (2, 1.5) vs. "starts at 8 and decreases 25% per step": decays faster? (the graph: 50% per step)
y = 4x vs. a table with x = 0, 1, 2, 3 and y = 1, 2, 4, 8: which is greater at x = 5? (32 > 20, the table)
"A ball thrown from 6 ft that reaches its greatest height, 22 ft, after 1 second" vs. h(t) = -16t² + 40t: higher maximum? (22 ft vs. 25 ft, the equation)
Procedure
Each group member takes two pairs, solves them alone, then explains them to the group
The group must agree on every answer and write the justification on a sticky note: "property, value, value, conclusion"
Groups trade their sticky notes with a neighbor group, which checks one answer by converting both cards to the same representation
Modification for Distance Learning
Post each pair on its own slide in a shared deck. Students type their justification in the speaker notes, and the teacher reviews one slide per group live.
2
Which Gym Membership?
20 minPairs
Pairs compare four gym memberships, each described in a different representation, and write a recommendation for three different customers. The context makes students compare initial values and rates of change and decide which comparison matters.
The Four Gyms
Gym 1 (verbal): "$30 to join, then $20 per month"
Gym 2 (equation): C(m) = 25m, where m is the number of months
Gym 3 (table): months 0, 3, 6, 9 and total cost $90, $132, $174, $216 ($14 per month after a $90 joining fee)
Gym 4 (graph, drawn on the handout): a line through (0, 50) and (6, 140) ($15 per month after a $50 fee)
Procedure
Pairs find the joining fee and the monthly rate of each gym and record them in the same units
They find the cheapest gym for someone who stays 3 months (Gym 2 at $75; Gym 1 costs $90), 6 months (Gym 4 at $140; Gyms 1 and 2 cost $150) and 12 months (Gym 4 at $230; Gym 3 costs $258)
Each pair writes a short recommendation for each customer, naming the property that decided it
Discussion Questions
Which representation made the monthly rate hardest to find? Why?
Gym 1 and Gym 2 cost the same after how many months? How can you see it in both representations?
Why is "lowest monthly rate" not the same as "cheapest"?
3
Build a Rival
20 minPairs
Each student receives one function in one representation and a target, such as "a larger maximum but the same y-intercept." The student builds a rival function in a different representation that meets the target, and the partner checks the claim.
Starter Cards and Sample Rivals
Given f(x) = -x² + 4x + 1 (maximum 5, y-intercept 1). Target: a table with a larger maximum and the same y-intercept. Sample: x = 0, 1, 2, 3, 4 and y = 1, 6, 7, 6, 1 (maximum 7)
Given a table with x = 0, 1, 2, 3 and y = 2, 6, 18, 54. Target: a verbal description of a linear function that is greater at x = 3 but smaller at x = 4. Sample: "starts at 10 and increases by 16 each step" (58 > 54 at x = 3; 74 < 162 at x = 4)
Given "a line with slope -2 and y-intercept 7." Target: an equation with the same x-intercept and a steeper slope. Sample: y = -4x + 14 (both reach 0 at x = 3.5)
Procedure
Students build their rival and write the property values of both functions on the back of the card
Partners swap, check both values in the new representation, and sign the card if the target is met
Pairs share one rival that surprised them with the class
Challenge Variation
Give a target that cannot be met, such as "a quadratic with a larger maximum than f(x) = -x² + 4x + 1 whose graph is below f everywhere," and ask students to explain why it is impossible.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: The Official Example, a Graph Against an Expression
Function f is given only by its graph, drawn to scale: a parabola through (0, 0), (1, 4.5), (2, 6), (3, 4.5) and (4, 0), with its highest point at (2, 6). Function g is given only by an expression. Completing the square shows that g has maximum 7, so g has the larger maximum even though its graph is not shown.
Diagram 2: Graph Bank for the Homework and Quiz
Four functions given only by their graphs, drawn to scale. Students read values, intercepts and turning points from these graphs to compare them with functions given in other ways in Quiz Questions 3, 4, 6 and 8 (Graphs A-D) and Homework Problems 2 and 6 (Graphs E and F). Read each property at a marked grid point: for example, a vertex is the dot where the graph turns.
04
Homework Assignment
~30 min
HSF.IF.C.9 Homework: Comparing Functions in Different Representations
Directions: Graphs E and F are in Diagram 2. For each comparison, write the property, the value for each function and your conclusion in a full sentence. Show how you found each value from its representation.
Part 1: Linear Functions (Problems 1-2)
Data plan A costs C(g) = 20 + 8g dollars per month for g gigabytes. Plan B's costs are in a table: 0 GB, $35; 2 GB, $45; 4 GB, $55; 6 GB, $65. Compare the base costs and the cost per gigabyte. For how many gigabytes do the plans cost the same?
Compare the function j in Graph E with f(x) = 0.5x + 5. Which has the greater y-intercept? Which has the greater rate of change? For which x-value are they equal, and what is the common value?
Part 2: Exponential and Quadratic Functions (Problems 3-5)
Investment A is $1,000 that earns 6% per year, compounded yearly. Investment B is in a table: year 0, $1,200; year 1, $1,250; year 2, $1,300; year 3, $1,350. Compare the initial values and the kind of growth. Which investment is worth more after 5 years, and which after 10 years?
Fountain A shoots water along h(x) = -0.2x² + 2x, in feet. Fountain B's water reaches a greatest height of 6 feet at a horizontal distance of 4 feet and lands 8 feet from the nozzle. Which fountain shoots higher, and which shoots farther?
Function r is described as "a parabola that opens upward, has its vertex at (-1, -5) and passes through (1, -1)." The quadratic u is given by a table: x = -2, -1, 0, 1, 2 and u(x) = 3, 0, -1, 0, 3. Compare their minimum values, axes of symmetry, y-intercepts and zeros.
Part 3: Four Representations (Problem 6)
Four functions: (a) f(x) = 5x - 2; (b) the function m in Graph F; (c) a table with x = 0, 1, 2 and y = 4, 6, 8; (d) "starts at 6 and decreases by 1 for each increase of 1 in x." Rank all four from greatest to least y-intercept. Then rank the three linear functions from greatest to least rate of change, and give the minimum value of m.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Reading Each Representation
Correct values from equations, graphs, tables and descriptions
One value read incorrectly
Several values wrong or missing
Comparing the Same Property
Every comparison uses the same property for both functions
One comparison mixes properties
Properties mixed or not named
Justification
Each conclusion states both values and the result in a sentence
Conclusions without both values
No justification
Context and Accuracy
Answers in context with units, all arithmetic correct
Minor arithmetic or unit errors
Answers do not fit the context
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Questions 3, 4, 6 and 8 use Graphs A, B, C and D from Diagram 2, so keep it open. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
f(x) = 4x - 7. A table for g gives x = 0, 1, 2, 3 and g(x) = -2, 1, 4, 7. Which function has the greater rate of change?
Answer: A
The slope of f is 4. In the table, g increases by 3 each time x increases by 1, so its rate is 3. Choice B uses the last table value, 7, as the rate. Choice C compares the y-intercepts, a different property.
Question 2 of 20 · Multiple Choice
f(x) = -x² + 8x - 10. Function g is a quadratic whose graph opens downward with vertex (3, 7). Which statement is true?
Answer: C
Completing the square: f(x) = -(x - 4)² + 6, so f has maximum 6 at x = 4. The vertex of g is its highest point, so g has maximum 7. Choice A reads the coefficient 8 as the maximum, and choice B uses the constant term.
Question 3 of 20 · Multiple Choice
Compare the function s in Graph D with h(x) = -2x² + 8x - 5. Which has the larger maximum?
Answer: B
Graph D turns at its highest point (-1, 5), so s has maximum 5. For h, x = -8/(2 · (-2)) = 2 and h(2) = -8 + 16 - 5 = 3. Choice A evaluates -2(2)² as +8. Choice C mixes up which function has which value.
Question 4 of 20 · Multiple Choice
Compare the function p in Graph A with the function k in this table: x = 0, 2, 4, 6 and k(x) = 1, 4, 7, 10. Which statement is true?
Answer: D
Graph A passes through (0, 3) and (2, 6), so p has slope 3/2 and y-intercept 3. In the table, k rises 3 for every 2 units of x, also a rate of 3/2, and k(0) = 1. Choice A reads the rate of k as 3 per unit, ignoring that the x-steps are 2.
Question 5 of 20 · Multiple Choice
Culture A is shown in a table: hour 0, 50 bacteria; hour 1, 100; hour 2, 200; hour 3, 400. Culture B starts with 120 bacteria and gains 60 bacteria each hour. Which statement is true?
Answer: A
A doubles each hour, so at hour 5 it has 50 · 25 = 1,600 bacteria. B grows by a constant 60, so at hour 1 it has 180 (A has 100) and at hour 5 it has 420. Choice B assumes the larger start wins, and choice C ignores that A is smaller at first.
Question 6 of 20 · Multiple Choice
Compare the function r in Graph C with m(x) = x² - 2x - 1. Which has the smaller minimum value?
Answer: C
Graph C turns at its lowest point (2, -3), so r has minimum -3. For m, completing the square gives (x - 1)² - 2, so its minimum is -2. Choice A uses m(0) = -1, the y-intercept, instead of the minimum.
Question 7 of 20 · Multiple Choice
Town F has 4,500 residents and grows 2% per year. Town G has population g(t) = 4,200(1.03)t after t years. Which statement is true?
Answer: B
Initial values: 4,500 for F and 4,200 for G, so F starts larger. Growth factors: 1.02 for F and 1.03 for G, so G grows at the larger rate, 3% per year. Choice A reads 4,200(1.03) as a larger start. Choice C compares 2% with 0.03 without writing both as percents.
Question 8 of 20 · Multiple Choice
Compare the function q in Graph B with the function w in this table: x = 0, 1, 2, 3 and w(x) = 3, 5, 7, 9. Which statement is true?
Answer: D
Graph B doubles: q(2) = 4 and q(4) = 16. The table grows by 2 per step, so w(2) = 7 and w(4) = 11. Exponential growth overtakes the linear function. Choice B assumes that the function greater at the start stays greater.
Question 9 of 20 · Multiple Choice
f(x) = (x - 5)(x + 1). A table for g gives x = -3, -2, -1, 0, 1, 2 and g(x) = 5, 0, -3, -4, -3, 0. Which function has its zeros farther apart?
Answer: A
f is 0 at x = 5 and x = -1, which are 6 units apart. The table shows g(-2) = 0 and g(2) = 0, which are 4 units apart. Choice B reads the zeros of g from the output 5 in the first row. Choice D counts rows instead of comparing a property.
Question 10 of 20 · Multiple Choice
Function f is linear: its graph crosses the y-axis at -4, and f increases by 3 for each increase of 1 in x. A table for k gives x = 0, 1, 2, 3 and k(x) = 2, 3.5, 5, 6.5. Compare f and k.
Answer: C
The slope of f is 3 and its y-intercept is -4. In the table, k increases by 1.5 per step and k(0) = 2. So f is steeper, and k starts higher. Choice A takes the last table value, 6.5, as if it showed the rate. Choice B compares -4 and 2 as if -4 were larger.
Question 11 of 20 · Multiple Choice
Ball A's height in feet is in a table: t = 0, 1, 2, 3, 4 seconds and h = 0, 48, 64, 48, 0. Ball B is thrown and reaches a greatest height of 70 feet. Which ball goes higher?
Answer: B
The heights in the table are symmetric about t = 2 (48 at t = 1 and t = 3), so t = 2 is the peak and Ball A's greatest height is 64 feet. Ball B reaches 70 feet. Choice A confuses the amount of data with the size of a value.
Question 12 of 20 · Multiple Choice
Compare the y-intercepts of f(x) = 3(0.5)x + 2 and the function h in this table: x = -2, 0, 2 and h(x) = 10, 4, 1.
Answer: D
The y-intercept of h is its value at x = 0, which the table gives as 4. For f, f(0) = 3 · 1 + 2 = 5. Choice A uses the first row of the table, where x = -2, not 0. Choice B forgets the + 2.
Question 13 of 20 · Multiple Choice
f(x) = 100x + 50. Quantity g starts at 1 when x = 0 and doubles each time x increases by 1. Which statement is true?
Answer: C
At x = 0, f(0) = 50 and g(0) = 1, so f starts larger. But g(x) = 2x is exponential: g(10) = 1,024 < 1,050 = f(10), and g(11) = 2,048 > 1,150 = f(11). A quantity that doubles eventually passes any linear function. Choice A ignores long-run behavior.
Question 14 of 20 · Multiple Choice
A table for h gives x = 0, 1, 2 and h(x) = 3, 4, 11. Compare the average rate of change of h and of f(x) = x² + x on the interval 0 ≤ x ≤ 2.
Answer: A
For h: (11 - 3)/(2 - 0) = 4. For f: f(0) = 0 and f(2) = 6, so (6 - 0)/2 = 3. Choice B uses the change in f without dividing by the change in x. Choice D uses the final values instead of rates.
Question 15 of 20 · Short Answer
Function n is described as "a parabola that opens upward with vertex (1, -5) that passes through (0, -3)." The quadratic v is given by a table: x = 0, 1, 2, 3, 4 and v(x) = 6, 3, 2, 3, 6. Compare their minimum values, axes of symmetry, y-intercepts and number of x-intercepts.
From the description, n(x) = a(x - 1)² - 5 and n(0) = a - 5 = -3, so a = 2: n(x) = 2(x - 1)² - 5. The table for v is symmetric about x = 2 (3 at x = 1 and x = 3), so v has axis x = 2 and minimum 2; n has axis x = 1 and minimum -5, the smaller minimum. y-intercepts: -3 for n and 6 for v. Because n goes below the x-axis and opens upward, it has two x-intercepts (at 1 ± √(5/2), about -0.58 and 2.58); v has minimum 2 > 0, so it has none.
Question 16 of 20 · Short Answer
Car A is worth V(t) = 24,000(0.88)t dollars after t years. Car B was bought for $30,000 and loses $2,500 in value each year. Compare the initial values, the loss in the first year, and the values after 5 years.
Initial values: $24,000 for A and $30,000 for B. First-year loss: A loses 12% of $24,000 = $2,880, B loses $2,500, so A loses more in the first year. After 5 years: A is worth 24,000(0.88)5 ≈ $12,666, B is worth 30,000 - 5(2,500) = $17,500. A loses a fixed percent and B a fixed amount.
Question 17 of 20 · Short Answer
A table for p gives x = 1, 2, 3, 4, 5 and p(x) = 2, 7, 10, 11, 10. Function q is q(x) = -x² + 6x + 3. Which function has the larger maximum? Justify both values.
For p, the outputs are symmetric about x = 4 (10 at x = 3 and at x = 5), so the maximum of p is 11, at x = 4. For q, completing the square gives q(x) = -(x - 3)² + 12, so the maximum of q is 12, at x = 3. q has the larger maximum.
Question 18 of 20 · Short Answer
Compare q(x) = 3x with the function g that starts at 5 when x = 0 and doubles for each increase of 1 in x. Compare their initial values and growth factors, and say which is greater at x = 2 and at x = 5.
q(0) = 1 and its growth factor is 3; g(x) = 5 · 2x starts at 5 and has growth factor 2. At x = 2: q(2) = 9 and g(2) = 20, so g is greater. At x = 5: q(5) = 243 and g(5) = 160, so q is greater: the larger growth factor wins in the long run, even though g starts higher.
Question 19 of 20 · Short Answer
Tank A holds V(t) = 500 - 12t liters of water after t minutes. Tank B is in a table: t = 0, 5, 10, 15 and volume 420, 360, 300, 240 liters. Compare their rates of change, and find which tank empties first.
Tank A drains 12 liters per minute. Tank B drains (420 - 360)/5 = 12 liters per minute, the same rate. Tank A starts with 500 liters and Tank B with 420, so Tank B empties first: at t = 420/12 = 35 minutes, while Tank A empties at 500/12 ≈ 41.7 minutes.
Question 20 of 20 · Short Answer
Function g is described as "a quadratic that opens downward, has zeros at -4 and 2, and passes through (0, 4)." Compare its maximum with the maximum of s(x) = -x² - 2x + 6.
From the description, g(x) = a(x + 4)(x - 2) and g(0) = -8a = 4, so a = -1/2. Its axis is halfway between the zeros, x = -1, and g(-1) = -1/2(3)(-3) = 4.5. Completing the square, s(x) = -(x + 1)² + 7, so s has the larger maximum, 7 compared with 4.5, even though both have the same axis of symmetry, x = -1.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does HSF.IF.C.9 mean?
HSF.IF.C.9 means students can compare a property of two functions when the functions are given in different ways: one as an equation, the other as a graph, a table or a description, for example. The official example asks which of two quadratic functions has the larger maximum when one is given by a graph and the other by an expression.
Is HSF.IF.C.9 Algebra 1 or Algebra 2?
It is usually taught in Algebra I with linear, quadratic and exponential functions, and it comes back in Algebra II with new function families such as polynomial, logarithmic and trigonometric functions. The skill is the same at every level: find the same property in both representations, then compare.
How is HSF.IF.C.9 different from 8.F.A.2?
The wording is almost the same, but 8.F.A.2 in grade 8 compares linear functions only, mostly their rates of change and initial values. HSF.IF.C.9 extends the comparison to any function type studied in high school, and to properties such as maximum and minimum values, zeros and long-run growth.
What properties can students compare?
Any property both functions have: intercepts, maximum or minimum values, the location of the vertex, zeros, the rate of change or average rate of change over an interval, the initial value and growth factor of an exponential, and which function is greater for large inputs.
How do you find the maximum of a quadratic from a table?
Look for symmetry in the outputs. If equal outputs appear at x = -3 and x = 1, the axis of symmetry is x = -1, and the output at x = -1 is the maximum or minimum. If the table has no symmetric pair, fit the equation from three points or say that the table alone does not settle the question.
What are common mistakes on HSF.IF.C.9 problems?
Common ones: comparing different properties (the slope of one function with the y-intercept of the other), reading the rate of change from a table without dividing by the x-step, taking the largest value in a table as the maximum without checking the pattern, and concluding that the function that starts larger stays larger.
Should students convert both functions to the same representation?
It is a good first strategy, especially converting both to a short table or an equation. As students get more confident, they should find the property directly in each representation, which is faster and is what the standard emphasizes. Converting is still a good way to check an answer.
How can a teacher check understanding of HSF.IF.C.9 quickly?
Give one function in two representations and ask a single comparison question with a sentence frame: "The ___ of f is ___, and the ___ of g is ___, so ___." Students who fill in two different properties, or leave one value blank, show exactly where the misunderstanding is.
Does HSF.IF.C.9 appear on the SAT?
Yes, in both the Algebra and Advanced Math domains of the digital SAT: questions give a function as a graph or table and another as an equation or description, and ask which has a greater value, slope or maximum. Reading the same property from both representations is the key step.
How does HSF.IF.C.9 connect to later topics?
Comparing representations leads into modeling: choosing between linear and exponential models from a table or a description (HSF.LE.A.1 and HSF.LE.A.3), building functions from a context (HSF.BF.A.1), and in statistics, comparing data sets and fitted models (HSS.ID.B.6).
07
Related Standards
6 standards
These standards connect to HSF.IF.C.9: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
8.F.A.2Prerequisite
Compare two functions each represented in a different way (grade 8, linear)
