HSF.IF.C.8: Rewriting Functions to Reveal Their Properties
In plain English: HSF.IF.C.8 is the Common Core functions standard that asks students to rewrite a function in an equivalent form that reveals a property. Factoring and completing the square show the zeros, maximum or minimum and symmetry of a quadratic, interpreted in context, and exponent rules show the growth or decay rate of an exponential function. It is usually taught in Algebra I and revisited in Algebra II.
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
a.Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
b.Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Common Core State Standards for Mathematics · Domain: Interpreting Functions (IF) · Cluster: Analyze functions using different representations Also written as HSF-IF.C.8 or F-IF.8 · Official standard
Students learn that one function can be written in several equivalent forms and that each form makes a different property easy to see. For quadratics (standard a), factoring reveals the zeros and completing the square reveals the maximum or minimum and the axis of symmetry. Students then explain what those features mean in a context, such as the height of a punted ball or the profit from a sale.
For exponentials (standard b), students use the properties of exponents to rewrite expressions such as (1.01)12t or 400(0.5)t/6 so that the exponent is t. The new base is the growth or decay factor per unit of t, and from it students read the percent rate of change and classify the function as growth or decay.
Learning Objectives
By the end of this lesson, students will be able to:
Factor a quadratic function to find its zeros and axis of symmetry
Complete the square in a quadratic function to find its maximum or minimum value and the vertex
Interpret zeros, extreme values and symmetry in terms of a context
Use the properties of exponents to rewrite exponential functions and identify the percent rate of change
Classify exponential functions as growth or decay from an equivalent form
Prior Knowledge Required
Students should already be comfortable with:
Factoring quadratic expressions HSA.SSE.A.2
Completing the square in a quadratic expression HSA.SSE.B.3
Properties of integer and rational exponents HSN.RN.A.2
Graphs of quadratic functions and their vertex HSF.IF.C.7
Write three expressions on the board and tell students all three define the same function:
Warm-Up Prompt
"f(x) = x² - 2x - 3, f(x) = (x - 3)(x + 1) and f(x) = (x - 1)² - 4. Check that they are equal by expanding. Then, for each one, name one fact about the graph you can read without doing any more work."
Collect facts on the board under each form. The standard form shows the y-intercept -3. The factored form shows the zeros 3 and -1. The vertex form shows the minimum -4 at x = 1 and the axis of symmetry x = 1. Ask: "Which form would you want if the question were about where the graph crosses the x-axis? About its lowest point?" That question is the whole lesson: choose the form that reveals the property you need.
Direct Instruction25 minutes
Part 1: Quadratics (standard a). Show the two processes and what each one reveals:
Factor ax² + bx + c into a(x - r)(x - s). The zeros are r and s, and the axis of symmetry is halfway between them, x = (r + s)/2.
Complete the square: factor a out of the x-terms, add and subtract the square of half the new x-coefficient, and write a(x - h)² + k. The vertex is (h, k): a minimum value k when a > 0, a maximum value k when a < 0.
Use symmetry: points with x-values equally far from x = h have equal outputs, so f(h - d) = f(h + d).
Interpret in context: zeros are the inputs where the quantity is 0 (the ball is on the ground, profit breaks even); the vertex gives the best or worst value and when it happens; values of the input outside the situation are rejected.
Work Examples 1-3 and use Diagram 1 to show the punt in all three forms. Part 2: Exponentials (standard b). Remind students that in y = a·bt, the base b is the growth factor: b > 1 means growth at rate b - 1, and 0 < b < 1 means decay at rate 1 - b. When the exponent is not just t, use the power rule (bm)n = bmn to rewrite the expression with exponent t, then read the rate. Work Example 4, the official example, and Example 5 with Diagram 2.
Factoring to show zeros and symmetry (part a)
Rewrite f(x) = x² - 8x + 12 in factored form and use it to find the zeros, the axis of symmetry and the minimum.
Completing the square to show the extreme value (part a)
Rewrite g(x) = 2x² + 12x + 7 by completing the square.
Equation: g(x) = 2(x² + 6x + 9) + 7 - 18 = 2(x + 3)² - 11: minimum -11 at x = -3, axis x = -3
Interpreting the forms in context (part a)
A punted ball has height h(t) = -5t² + 20t meters after t seconds (using 5 for half the acceleration of gravity, rounded). Write it in factored and vertex form and interpret each.
Equation: h(t) = -5t(t - 4) = -5(t - 2)² + 20: in the air from t = 0 to t = 4 s; highest point 20 m at t = 2 s; equal heights at t = 1 and t = 3
Official example: percent rate of change (part b)
Identify the percent rate of change of y = (1.02)t, y = (0.97)t, y = (1.01)12t and y = (1.2)t/10, and classify each as growth or decay.
Equation: 2% growth; 3% decay; (1.0112)t ≈ (1.1268)t, about 12.68% growth per unit of t; (1.21/10)t ≈ (1.0184)t, about 1.84% growth per unit of t (20% per 10 units)
Rewriting a half-life model (part b)
A 400 mg dose of a medicine with a 6-hour half-life leaves A(t) = 400(0.5)t/6 mg in the body after t hours. What is the hourly rate?
Equation: A(t) = 400((0.5)1/6)t ≈ 400(0.891)t: about 10.9% of the medicine is removed each hour
Stress one idea about Example 4: (1.01)12t and (1.1268)t (rounded) are the same function, so "1% per month" and "about 12.68% per year" describe the same growth. The rewritten form does not change the function; it reveals a different property of it.
Guided Practice15-20 minutes
Pairs work three problems, and you stop after each one to compare forms: (1) factor f(x) = x² + 4x - 21 = (x + 7)(x - 3) to find the zeros -7 and 3, the axis x = -2 and the minimum f(-2) = -25; (2) complete the square for f(x) = -x² + 6x + 1 = -(x - 3)² + 10 to find the maximum 10 at x = 3; (3) rewrite P(t) = 1000(1.005)12t, where t is in years, as about 1000(1.0617)t and state the monthly rate (0.5%) and the yearly rate (about 6.17%). Listen for these errors: taking the zeros of (x + 7)(x - 3) as 7 and -3, forgetting to multiply the added square by the leading coefficient -1, and multiplying 0.5% by 12 instead of raising 1.005 to the 12th power.
Independent Practice15 minutes
Students work alone on four problems: (1) f(x) = x² - 10x + 16: factor to (x - 2)(x - 8) and give the zeros 2 and 8 and the vertex (5, -9); (2) f(x) = 3x² - 12x + 5: complete the square to 3(x - 2)² - 7, minimum -7; (3) y = (0.9)t/2: rewrite as ((0.9)1/2)t ≈ (0.9487)t, a decay of about 5.13% per unit of t and 10% per 2 units; (4) y = 3t + 2: rewrite as 9·3t to show the initial value 9 and tripling. Students write one sentence for each problem naming the property the new form reveals.
Closure5-10 minutes
Exit ticket: (1) f(x) = (x + 2)(x - 4) = (x - 1)² - 9. Which form gives the zeros, and which gives the minimum? (Factored: -2 and 4; vertex: -9 at x = 1.) (2) Is y = 5(0.96)t growth or decay, and at what percent rate? (Decay, 4% per unit of t.) (3) In one sentence, explain why rewriting an expression does not change the graph.
Differentiation Strategies
For Struggling Students
Give a three-column organizer (standard, factored, vertex) with a row for "what this form shows" and have students fill it in for every quadratic
Use an area model for completing the square, starting with a = 1 before factoring out a leading coefficient
For exponentials, give a two-step routine: first make the exponent t with (bm)n = bmn, then compare the new base with 1
For Advanced Students
Ask students to complete the square on f(x) = ax² + bx + c in general and show that the vertex is at x = -b/(2a)
Ask which is the better savings offer: 3% per year compounded yearly, or 0.25% per month, and justify with equivalent forms
Ask students to find the doubling time of y = (1.1268)t by trying values of t, and connect it to (1.01)12t
Assessment Guidance
What to Look For
Students should name the property before they rewrite: "I want the maximum, so I will complete the square." Check that zeros come with the right signs, that the added square is multiplied by the leading coefficient when completing the square, and that context answers include units and reject inputs that make no sense, such as negative time. For exponentials, look for the power rule used correctly and for the rate stated with its time unit: "about 10.9% per hour," not only "10.9%."
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Classroom Activities
3 Activities
1
Three Forms Card Sort
20 minGroups of 3
Each group gets 18 cards: six quadratic functions, each written in standard, factored and vertex form. Groups sort the cards into six matching triples, then write the zeros, the vertex and the axis of symmetry on a feature card for each triple, naming which card revealed each feature.
The Six Triples (teacher key)
x² - 2x - 15 = (x - 5)(x + 3) = (x - 1)² - 16: zeros 5 and -3, minimum -16 at x = 1
x² + 6x + 5 = (x + 1)(x + 5) = (x + 3)² - 4: zeros -1 and -5, minimum -4 at x = -3
-x² + 4x + 12 = -(x - 6)(x + 2) = -(x - 2)² + 16: zeros 6 and -2, maximum 16 at x = 2
2x² - 8x - 10 = 2(x - 5)(x + 1) = 2(x - 2)² - 18: zeros 5 and -1, minimum -18 at x = 2
-2x² - 4x + 6 = -2(x + 3)(x - 1) = -2(x + 1)² + 8: zeros -3 and 1, maximum 8 at x = -1
Procedure
Groups match cards by expanding or factoring; every match must be checked by expanding one card of the triple
For each triple, the group checks that the axis of symmetry from the vertex form is the midpoint of the zeros from the factored form
Groups post their six feature cards, then walk to another group's poster and check one triple
Modification for Distance Learning
Put the 18 cards on a shared slide as draggable text boxes. Groups drag each triple into a row and type the features in a fourth column.
2
What Does the Form Say About the Situation?
20 minPairs
Pairs rewrite two context functions in factored and vertex form and write a sentence for each feature, with units. This activity targets the second half of standard a: interpreting zeros, extreme values and symmetry in terms of a context.
The Two Situations
A drinking fountain shoots water along the path h(x) = -0.5x² + 3x, where h is the height in inches above the nozzle and x the horizontal distance in inches. Forms: -0.5x(x - 6) and -0.5(x - 3)² + 4.5. The water leaves the nozzle at x = 0, lands back at nozzle height 6 inches away, and peaks 4.5 inches above the nozzle, 3 inches out
A school club sells bracelets. At a price of p dollars its profit is P(p) = -2p² + 120p - 1000 dollars. Forms: -2(p - 10)(p - 50) and -2(p - 30)² + 800. The club breaks even at $10 and at $50, and the best price is $30, for a profit of $800
Procedure
Partner A factors, Partner B completes the square; then they swap and check each other by expanding
Together they write three sentences per situation: what the zeros mean, what the vertex means, and one pair of inputs with equal outputs (for example, prices of $20 and $40 give the same profit)
Pairs mark any input values that do not make sense in the situation, such as negative prices
Discussion Questions
Why do prices of $20 and $40 give the same profit? What does that have to do with the axis of symmetry?
Which form would a club treasurer want, and why?
What would a negative value of h mean for the fountain?
3
Growth Rate Rewrite Relay
20 minGroups of 4
Teams race through 6 exponential cards. For each card, one student rewrites the expression so the exponent is t, the next states the growth or decay factor per unit of t, the third gives the percent rate and classifies it, and the fourth checks with a calculator. This practices standard b: using properties of exponents to interpret exponential functions.
Cards (with answers)
y = 1000(1.015)4t (1.5% per quarter): ≈ 1000(1.0614)t, about 6.14% growth per year
y = 60(0.5)t/3: ≈ 60(0.7937)t, about 20.6% decay per unit of t
y = 5(1.1)t/5: ≈ 5(1.0192)t, about 1.92% growth per unit of t
y = 32t: = 9t, 800% growth per unit of t
y = 400(1.25)-t: = 400(0.8)t, 20% decay per unit of t
y = 2t - 1: = 0.5·2t, initial value 0.5, 100% growth per unit of t
Procedure
Each team member does one step and passes the card; roles rotate after every card
A card is finished only when the calculator check matches to four decimal places
The team writes the exponent rule used on each card: (bm)n = bmn, b-n = (1/b)n or bm + n = bmbn
Challenge Variation
Teams write their own card: a function whose exponent is not t but whose rate per unit of t is exactly 44% growth (for example, y = (1.2)2t), and trade with another team.
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Diagrams & Visual Aids
2 diagrams
Diagram 1: One Quadratic Function, Three Forms
The punt model h(t) = -5t² + 20t drawn to scale for 0 ≤ t ≤ 4. The factored form -5t(t - 4) gives the zeros t = 0 and t = 4 (kick and landing). The completed-square form -5(t - 2)² + 20 gives the maximum height 20 m at t = 2 s. The points (1, 15) and (3, 15) show the symmetry about the axis t = 2.
Diagram 2: Rewriting an Exponential to Read Its Rate
A(t) = 400(0.5)t/6 drawn to scale: the amount halves every 6 hours (400, 200, 100, 50, 25 mg). The power rule turns it into about 400(0.891)t, which shows that about 10.9% of the medicine is removed each hour. Both forms are the same function.
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Homework Assignment
~30 min
HSF.IF.C.8 Homework: Equivalent Forms That Reveal Properties
Directions: For each problem, first name the property you are asked for and the form that reveals it, then rewrite. Show every step. In context problems, answer in a full sentence with units and say whether any value should be rejected.
Part 1: Factoring and Completing the Square (Problems 1-2)
Factor f(x) = x² + 2x - 24. Use the factored form to find the zeros and the axis of symmetry, then find the minimum value of f.
Complete the square to write g(x) = 3x² + 18x + 20 in the form a(x - h)² + k. State the vertex, the minimum value and the axis of symmetry, and find the exact zeros.
Part 2: Quadratics in Context (Problems 3-4)
A model rocket is launched from the ground, and its height in feet after t seconds is h(t) = -16t² + 96t. Write h in factored form and in vertex form. What do the zeros, the vertex and the axis of symmetry tell you about the flight? At what two times is the rocket 80 feet high?
A theater finds that at a ticket price of p dollars, its revenue from one show is R(p) = -30p² + 1200p dollars. Rewrite R in two ways to find (a) the prices that bring in no revenue and (b) the price that brings in the most revenue, and the amount. Explain why a price of $15 and a price of $25 bring in the same revenue.
Part 3: Exponential Functions (Problems 5-6)
For each function, use properties of exponents to write it as a·bt, classify it as growth or decay, and give the percent rate of change per unit of t (round to two decimal places): (a) y = 450(1.035)t (b) y = 90(0.8)2t (c) y = 12(1.5)t/4 (d) y = 20(2)-t.
A used car is worth V(t) = 28,000(0.85)t dollars t years after purchase. Explain what 28,000 and 0.85 mean. Then rewrite V to show the monthly factor and the percent the car loses each month.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Choosing the Form
Names the property and the form that reveals it every time
Correct form, property not named
Form missing or wrong
Rewriting
Factoring, completing the square and exponent rules all correct
One algebra or exponent error
Several errors or no rewriting
Interpretation in Context
Zeros, extremes and symmetry explained with units; bad values rejected
Interpretation missing units or one feature
No interpretation
Rates of Change
Growth or decay and percent rate with its time unit all correct
Classification right, rate wrong or no unit
Missing or incorrect
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Quiz: 20 Questions
Interactive, with answers
Instructions
Use a scientific calculator for the exponential questions. 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
Which equivalent form of f(x) = x² - 4x - 12 shows its zeros directly?
Answer: B
In factored form, f(x) = 0 when x - 6 = 0 or x + 2 = 0, so the zeros are 6 and -2. Choice A is also equivalent, but it shows the minimum -16 at x = 2, not the zeros. Choice C is equivalent but not fully factored. Choice D has the signs reversed: it expands to x² + 4x - 12.
Question 2 of 20 · Multiple Choice
Complete the square: what is the minimum value of f(x) = x² + 10x + 21?
Answer: C
x² + 10x + 21 = (x² + 10x + 25) - 25 + 21 = (x + 5)² - 4. Since (x + 5)² ≥ 0, the minimum value is -4, at x = -5. Choice A is the y-intercept. Choice B is the x-value where the minimum happens, not the minimum value. Choice D has the wrong sign.
Question 3 of 20 · Multiple Choice
What is the axis of symmetry of the graph of f(x) = -(x - 3)(x + 7)?
Answer: A
The zeros are 3 and -7, and the axis of symmetry is halfway between them: x = (3 + (-7))/2 = -2. Choice B takes the zeros as -3 and 7, a sign error. Choice C is half the distance between the zeros, not the midpoint. Choice D multiplies the constants.
Question 4 of 20 · Multiple Choice
Which is f(x) = 2x² - 12x + 11 written in vertex form?
Answer: D
Factor 2 from the x-terms: 2(x² - 6x) + 11. Add and subtract 9 inside: 2(x² - 6x + 9) - 18 + 11 = 2(x - 3)² - 7. Choice A subtracts 9 instead of 2 · 9 = 18, a common error when the leading coefficient is not 1. Choice C has the wrong sign inside the square, and choice B uses half of 12 without factoring out the 2.
Question 5 of 20 · Multiple Choice
A store models its weekly profit from a phone case sold at x dollars as P(x) = -5(x - 40)² + 2000 dollars. What does the vertex form tell the store?
Answer: A
The vertex is (40, 2000). Because the coefficient -5 is negative, P has a maximum there: the best price is $40 and the most profit is $2,000 per week. Choice B swaps the input and output. Choice C reads the constant as the y-intercept, but P(0) = -8000 + 2000 = -6000. Choice D calls the vertex a minimum.
Question 6 of 20 · Multiple Choice
A diver's height above the water after t seconds is h(t) = -5t² + 5t + 10 = -5(t - 2)(t + 1) meters. When does the diver reach the water?
Answer: B
h(t) = 0 when t = 2 or t = -1. Time before the dive starts (t = -1) does not fit the situation, so the diver reaches the water at t = 2 seconds. Choice A keeps the zero that makes no sense in context. Choice C comes from a sign error in the factor (t + 1). Choice D reads the constant term 10, which is the height of the platform.
Question 7 of 20 · Multiple Choice
A ball's height is h(t) = -16t² + 48t + 4 feet after t seconds. Its height at t = 0.5 seconds equals its height at which other time?
Answer: D
Completing the square gives h(t) = -16(t - 1.5)² + 40, so the axis of symmetry is t = 1.5. The time 0.5 is 1 second before the axis, so the matching time is 1 second after it: t = 2.5. Check: h(0.5) = h(2.5) = 24 feet. Choice C doubles 1.5 instead of reflecting 0.5 across it.
Question 8 of 20 · Multiple Choice
Classify y = 300(0.92)t and give its percent rate of change.
Answer: C
The base 0.92 is less than 1, so this is decay, and each unit of t keeps 92% of the amount: it loses 1 - 0.92 = 0.08 = 8%. Choice B confuses the part that remains with the part that is lost. Choice D writes 0.08 as a percent without multiplying by 100.
Question 9 of 20 · Multiple Choice
What is the percent rate of change per unit of t for y = (1.03)4t?
Answer: A
By the power rule, (1.03)4t = (1.034)t ≈ (1.1255)t, so the rate is about 12.55% growth per unit of t. Choice B is the rate per quarter of a unit. Choice C multiplies 3% by 4, which ignores compounding.
Question 10 of 20 · Multiple Choice
A substance decays by y = 100(0.5)t/8, where t is in years. About what percent of the substance is lost each year?
Answer: D
Rewrite: (0.5)t/8 = ((0.5)1/8)t ≈ (0.917)t. Each year about 91.7% remains, so about 8.3% is lost. Choice A is the loss every 8 years. Choice B divides 50% by 8, which ignores compounding. Choice C is the part that remains, not the part lost.
Question 11 of 20 · Multiple Choice
Which expression is equivalent to 2t + 3 and shows the value at t = 0?
Answer: A
By the product rule, 2t + 3 = 2t · 23 = 8·2t: the value at t = 0 is 8 and the amount doubles each unit. Choice B adds instead of multiplying. Choice C multiplies 2 · 3 instead of computing 23. Choice D moves the 3 into the exponent as a factor.
Question 12 of 20 · Multiple Choice
What is the percent rate of change per unit of t for y = (1.44)t/2?
Answer: B
(1.44)t/2 = ((1.44)1/2)t = (1.2)t, so the rate is 20% growth per unit of t. Choice A is the rate per 2 units. Choice C divides 44% by 2, which ignores compounding: 1.22² = 1.4884, not 1.44.
Question 13 of 20 · Multiple Choice
A student writes f(x) = x² - 6x + 5 as (x - 3)² + 5 and says the minimum value is 5. What went wrong?
Answer: D
(x - 3)² = x² - 6x + 9, so the student added 9 without taking it away. The correct form is (x - 3)² - 4, with minimum -4 at x = 3. Check: f(3) = 9 - 18 + 5 = -4. Choice B changes a correct part of the work. Choice C is false: completing the square is exactly the process that shows the extreme value.
Question 14 of 20 · Multiple Choice
A savings account grows by A(t) = 2000(1.004)12t dollars, where t is in years and the account earns 0.4% per month. What is the yearly percent growth?
Answer: C
(1.004)12t = (1.00412)t ≈ (1.0491)t, so the account grows about 4.91% per year. Choice A is the monthly rate. Choice B multiplies 0.4% by 12, which ignores the interest earned on interest. Choice D is off by a factor of 10.
Question 15 of 20 · Short Answer
Factor f(x) = x² - 3x - 28. Use the factored form to give the zeros and the axis of symmetry, then find the vertex.
f(x) = (x - 7)(x + 4), so the zeros are 7 and -4. The axis is halfway between them: x = (7 + (-4))/2 = 1.5. f(1.5) = 2.25 - 4.5 - 28 = -30.25, so the vertex is (1.5, -30.25), a minimum.
Question 16 of 20 · Short Answer
Complete the square to find the maximum value of f(x) = -x² + 8x - 7. Then use factoring to find its zeros.
f(x) = -(x² - 8x) - 7 = -(x² - 8x + 16) + 16 - 7 = -(x - 4)² + 9, so the maximum is 9 at x = 4. Factoring: f(x) = -(x² - 8x + 7) = -(x - 1)(x - 7), so the zeros are 1 and 7, symmetric about x = 4.
Question 17 of 20 · Short Answer
A gardener has 60 meters of fence for a rectangular garden. With width w meters, the area is A(w) = w(30 - w) square meters. Rewrite A to find the width that gives the largest area, and explain what the zeros of A mean.
Completing the square: A(w) = -w² + 30w = -(w - 15)² + 225. The largest area is 225 m², when w = 15 m (the garden is a 15 m by 15 m square). The factored form w(30 - w) has zeros 0 and 30: a width of 0 m or 30 m leaves a length of 30 m or 0 m, so the "rectangle" has no area. Only widths between 0 and 30 make sense.
Question 18 of 20 · Short Answer
A town's population is P(t) = 12,000(1.1)t/2, where t is in years. Rewrite P to find the yearly growth rate, and explain what 1.1 means in the original form.
P(t) = 12,000((1.1)1/2)t ≈ 12,000(1.0488)t, so the population grows about 4.88% per year. In the original form, 1.1 is the growth factor for every 2 years: the population grows 10% every 2 years, which is less than 5% per year because the growth compounds.
Question 19 of 20 · Short Answer
Show that y = 9t/2 and y = 3t are the same function, and give its percent rate of change per unit of t.
9t/2 = (32)t/2 = 32 · t/2 = 3t, by the power rule. The growth factor per unit of t is 3, so the function grows by 3 - 1 = 2 = 200% per unit of t (it triples).
Question 20 of 20 · Short Answer
The same function is written as f(x) = (x + 1)(x - 5) and as f(x) = (x - 2)² - 9. Show that the two forms are equivalent, and list the features each form reveals.
Both expand to x² - 4x - 5: (x + 1)(x - 5) = x² - 5x + x - 5, and (x - 2)² - 9 = x² - 4x + 4 - 9. The factored form reveals the zeros -1 and 5. The vertex form reveals the minimum -9 at x = 2 and the axis of symmetry x = 2, which is the midpoint of -1 and 5. The standard form reveals the y-intercept -5.
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does HSF.IF.C.8 mean?
HSF.IF.C.8 means students rewrite a function in an equivalent form to reveal and explain a property. Part a uses factoring and completing the square to show the zeros, extreme values and symmetry of a quadratic and interpret them in context. Part b uses properties of exponents to interpret exponential functions, such as finding the percent rate of change.
Is HSF.IF.C.8 Algebra 1 or Algebra 2?
It is usually taught in Algebra I and extended in Algebra II. Factoring and completing the square for quadratics, and simple growth and decay rates, are Algebra I topics. Rewrites such as (1.01)12t ≈ (1.1268)t or half-life models often come back in Algebra II with logarithms.
How is HSF.IF.C.8 different from HSA.SSE.B.3?
HSA.SSE.B.3 is about rewriting an expression to reveal properties of the quantity it represents. HSF.IF.C.8 applies the same algebra to a function and its graph: the zeros become x-intercepts, the extreme value becomes the vertex, and the students explain the symmetry of the graph and what the features mean in a context.
When should students factor and when should they complete the square?
Factor when the question is about zeros and the quadratic factors over the integers. Complete the square when the question is about the maximum or minimum, the vertex or the axis of symmetry, or when the quadratic does not factor. Both forms give the axis of symmetry: halfway between the zeros, or x = h in a(x - h)² + k.
How do you find the percent rate of change of an exponential function?
Write the function as y = a·bt with the exponent exactly t. If b > 1, it grows at the rate b - 1 per unit of t; if 0 < b < 1, it decays at the rate 1 - b. For example, y = (0.97)t decays 3% per unit of t, and y = (1.02)t grows 2%. When the exponent is something like 12t or t/10, use (bm)n = bmn first.
Why is (1.01)12t not 12% growth per year?
Because growth compounds. (1.01)12t = (1.0112)t ≈ (1.1268)t, so 1% per month is about 12.68% per year. Each month the 1% applies to a slightly larger amount than the month before, so the yearly total is a little more than 12 × 1%.
What are common mistakes with completing the square in a function?
Common ones: adding the square without subtracting it, so the function changes; forgetting to multiply the added square by the leading coefficient after factoring it out, as in 3(x² + 4x + 4), which adds 12, not 4; and reading the vertex of a(x - h)² + k as (-h, k). Checking by expanding the new form catches all three.
What does "interpret in terms of a context" look like for a quadratic?
It means a sentence with units that says what the feature means in the situation. For a height model: the zeros are the times the object is on the ground, the vertex gives the greatest height and when it happens, and symmetry means it is at the same height at equal times before and after the peak. Students should also reject inputs that do not fit, such as negative time.
Does HSF.IF.C.8 show up on the SAT?
Yes, the skills are part of the Advanced Math domain of the digital SAT: questions ask which equivalent form of a quadratic shows its vertex or zeros, or what a constant in an exponential model means. Knowing which form reveals which property saves time on those questions.
How does HSF.IF.C.8 connect to later topics?
Completing the square returns in finding the center and radius of a circle (HSG.GPE.A.1) and in deriving the quadratic formula (HSA.REI.B.4). Rewriting exponential rates leads to interpreting parameters in exponential models (HSF.LE.B.5), to compound interest, and to logarithms in Algebra II.
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Related Standards
6 standards
These standards connect to HSF.IF.C.8: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
HSA.SSE.B.3Prerequisite
Choose and produce an equivalent form of an expression to reveal properties
