Grade by Grade
A. Characteristics of Functions · Quadratic Functions

Quadratic Functions: Unit Review

A2.1 A2.2 A2.3 A2.4 A2.5 A3.2About 40 minutes
Learning goals
  • I can review every lesson in the Quadratic Functions unit.
  • I can find which lessons I need to revisit.

Let's review!

In this unit you wrote quadratic functions in three forms and used function notation. You found maximum and minimum values by completing the square, by partial factoring and with x = −b/(2a). You simplified radicals so you could give exact zeros, counted zeros with graphs, factoring and the discriminant, and used the discriminant in reverse to find unknown coefficients. You studied families of quadratic functions with the same zeros and built equations from zeros and a point. You solved revenue, profit, area and projectile problems, and found where lines meet parabolas, including tangent lines.

This review has a few questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do one or two sections at a time. For every question, first decide which tool you need (vertex, zeros, discriminant, family, system), then solve. Give exact answers unless the question asks you to round. For every question you get wrong, write down why (sign error? wrong form? forgot a? wrong side of the zeros?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

Max or mincomplete the squareor x = −b/(2a)a > 0: min; a < 0: maxHow many zeros?D = b² − 4acD > 0: 2, D = 0: 1D < 0: noneExact zerosquadratic formulasimplify √ anddivide every termSame zeros r, sf(x) = a(x − r)(x − s)one more pointgives aLine meets parabolaset f(x) = g(x)solve the quadraticD = 0: tangentWord problemsdefine, write f(x),restrict the domain,answer with units
The six big tools of the unit. Most questions use one or two of them.

Quadratic Functions in Three Forms

1

For f(x) = −(x − 3)(x + 1), state the zeros, the vertex and the maximum value.

2

Write f(x) = 2(x + 1)² − 8 in factored form.

3

For g(x) = x² − 6x + 8, find g(−1).

Maximum and Minimum Values by Completing the Square

1

Write f(x) = x² + 7x + 10 in vertex form and state the minimum value.

2

Find the maximum value of f(x) = −3x² + 18x − 20.

3

Find the minimum value of f(x) = 0.5x² − 4x + 3.

Maximum and Minimum Values: Partial Factoring and Symmetry

1

Use partial factoring to find the minimum value of f(x) = 2x² + 8x − 5.

2

Find the exact maximum value of f(x) = −x² + 5x + 1.

3

A parabola has zeros 3 and 11. What is its axis of symmetry?

Exact Answers: Simplifying and Operating with Radicals

1

Simplify √45.

2

Simplify √27 + √75.

3

Expand and simplify (3 + √2)(1 − √2).

4

Simplify (10 − √50)/5.

Exact Zeros: Solving Quadratic Equations with Radicals

1

Solve x² − 8x + 13 = 0 exactly.

2

Solve 2x² + 4x − 3 = 0 exactly.

3

Find the exact zeros of f(x) = 3(x − 2)² − 15.

How Many Zeros? Graphs, Factoring and the Discriminant

1

How many zeros does f(x) = 5x² − 2x + 1 have?

2

How many zeros does f(x) = −4(x + 3)² + 1 have? Explain without calculating.

3

f(x) = (x − 1)² − 4. How many zeros does f(x) + 4 have?

Using the Discriminant: Unknown Coefficients and "Can It Reach?"

1

Find k so that x² + kx + 49 has exactly one zero.

2

For what values of c does 2x² + 8x + c have no zeros?

3

h(t) = −5t² + 10t + 2 is a ball's height in metres. Does it reach 8 m?

Families of Quadratic Functions with the Same Zeros

1

Write the family with zeros −4 and 2, and find the member through (0, −16).

2

Find the member of the same family with a maximum value of 18.

3

How is y = ½(x + 4)(x − 2) related to y = (x + 4)(x − 2)?

Finding the Equation from Zeros and a Point

1

Find the quadratic function with zeros 3 ± √7 that passes through (3, 14).

2

Find the quadratic function with zeros −1/3 and 2 and y-intercept 4. Write it in standard form.

3

A parabola touches the x-axis only at x = −2 and has y-intercept 12. Find its equation.

Real-World Problems: Revenue and Profit

1

A club sells 200 tickets at $10. Each $1 increase loses 10 sales. Find the price for maximum revenue and the maximum revenue.

2

P(x) = −x² + 40x − 300. Find the break-even values and the maximum profit.

3

For the same P(x), when is the profit at least 75?

Real-World Problems: Area, Projectiles and Numbers

1

A rectangular field beside a river is fenced on three sides with 120 m of fence. Find the maximum area.

2

A ball's height is h(t) = −4.9t² + 9.8t + 1 metres. Find its maximum height and when it lands (to the nearest hundredth).

3

State a sensible domain and range for that ball.

Where a Line Meets a Parabola

1

Find the intersection points of f(x) = x² − 2x − 3 and g(x) = x + 1.

2

Solve f(x) = x² + 4, g(x) = 4x.

3

Do f(x) = −x² + 1 and g(x) = x + 3 intersect?

−2−1012345−5−4−3−2−1012345678xy(4, 5)vertex (1, −4)(−1, 0)g(x) = x + 1f(x) = x² − 2x − 3
A linear-quadratic system from the review: f(x) = x² − 2x − 3 and g(x) = x + 1 meet at (−1, 0) and (4, 5). Can you see why (−1, 0) is also a zero of f?

Lines That Touch, Cross or Miss: Linear-Quadratic Problems

1

Find b so that y = 3x + b is tangent to y = x² + x, and the point of tangency.

2

For which b do lines y = −x + b meet f(x) = x(4 − x) twice?

3

Find m so that y = mx + 2 is tangent to y = x² + 6.

Just for fun

You can now find the highest point of a ball's flight, the best price for snowshoe rentals and the exact spot where a snowball hits a hill. Not bad for a function whose biggest exponent is 2.

Check your understanding

Quiz

1.

What is the vertex of f(x) = 3(x − 2)² − 5?

2.

What is the minimum value of f(x) = x² − 10x + 4?

3.

Which is √98 in simplest form?

4.

What are the solutions of x² − 2x − 2 = 0?

5.

How many zeros does f(x) = 4x² − 4x + 1 have?

6.

Which equation represents the family of quadratic functions with zeros 1 and 6?

7.

For which values of k does x² + kx + 4 have no zeros?

8.

R(x) = (20 + x)(100 − 4x). What is the maximum revenue?

9.

Where do y = x² and y = 3x − 2 intersect?

10.

A ball's height is h(t) = −5t² + 20t + 1 metres. What is its maximum height?

For parents and teachers

Use the review to find what to revisit:

  • Reading the vertex with the wrong sign, or mixing up forms → Lesson 1 (Quadratic Functions in Three Forms).
  • Errors completing the square, especially when a ≠ 1 or b is odd → Lesson 2 (Completing the Square).
  • Forgetting to substitute after finding x = −b/(2a), or partial-factoring mistakes → Lesson 3 (Partial Factoring and Symmetry).
  • Radicals not fully simplified, or unlike radicals added → Lesson 4 (Simplifying and Operating with Radicals).
  • Quadratic formula answers not simplified, or only part of the numerator divided → Lesson 5 (Exact Zeros).
  • Wrong number of zeros, or sign errors in b² − 4ac → Lesson 6 (How Many Zeros?).
  • Solving k² > 16 as k > 4 only, or not rearranging a "can it reach" question → Lesson 7 (Using the Discriminant).
  • Sign errors in factors from zeros, or using a zero as the extra point → Lessons 8 and 9 (Families; Equation from Zeros and a Point).
  • Units, the wrong side of the break-even values, or answering with the number of increases instead of the price → Lesson 10 (Revenue and Profit).
  • Fencing set-up, negative times kept, or missing domain restrictions → Lesson 11 (Area, Projectiles and Numbers).
  • Forgetting the y-coordinates, or subtracting expressions incorrectly → Lesson 12 (Where a Line Meets a Parabola).
  • Tangent-line set-ups → Lesson 13 (Lines That Touch, Cross or Miss).

Next in MCR3U is the unit Exponential Functions. In Grade 12, Advanced Functions (MHF4U) extends this work to polynomial functions of higher degree and to polynomial inequalities.