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A. Quadratic Functions · Quadratic Functions

Quadratic Functions: Unit Review

A2.1 A2.2 A2.3 A2.4 A2.5 A2.6 A2.7 A2.8 A2.9 A2.10 A2.11 A3.1 A3.2 A3.3About 40 minutes
Learning goals
  • I can explain functions, function notation, domain and range, including restrictions in context.
  • I can describe and sketch transformations of f(x) = x².
  • I can move between the standard, vertex and factored forms and sketch from each one.
  • I can model data with a quadratic function and solve real-world problems.

Words to know

function
A relation in which each input has exactly one output.
domain / range
The set of inputs / the set of outputs.
vertex form
f(x) = a(x − h)² + k, with vertex (h, k).
standard form
f(x) = ax² + bx + c, with y-intercept c.
factored form
f(x) = a(x − r)(x − s), with zeros r and s.
quadratic regression
A technology-generated quadratic curve of best fit.

Let's review!

In this unit you learned what a function is and how to use function notation, domain and range, including restrictions in real situations. You investigated how a, h and k transform f(x) = x², sketched parabolas by transformations, and moved between the standard, vertex and factored forms by expanding, completing the square and factoring. You sketched from every form. Finally you collected quadratic data, found models by hand and with technology, and solved problems about heights, arches, areas, revenue and profit.

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

How to use this review

Do one or two sections at a time. Write full solutions, and sketch a graph whenever the question is about one. Keep a calculator handy for the quadratic formula, and Desmos if you want to check graphs. For each question you get wrong, write down why (sign error? wrong form? forgot to multiply by a? kept a negative time?) and go back to that lesson's worked examples. Finish with the 10-question quiz.

standard formf(x) = ax² + bx + cy-intercept ch = −b ÷ (2a)partial factoringvertex formf(x) = a(x − h)² + kvertex (h, k), axis x = hmax or min k, rangetransformations of x²factored formf(x) = a(x − r)(x − s)zeros r and saxis x = (r + s) ÷ 2sketch from the zerosstandard → vertex:complete the squarevertex or factored → standard:expandstandard → factored:factor (if real zeros exist)a is the same in all three forms: direction and stretch
The three forms of a quadratic function and what each one shows at a glance. Expanding, factoring and completing the square move you between them.

Relations and Functions: What Makes a Function?

1

Is {(2, 5), (3, 5), (2, 7)} a function? Explain.

2

Is y = 3x² − 1 a function? Is x² + y² = 4?

3

Does parabola P in the graph below pass the vertical line test?

xy−5−4−3−2−101234567−3−2−1012345678910parabola P
Parabola P, used in several review questions. The marked points are (−3, 0), (0, 7.5), (1, 8), (2, 7.5) and (5, 0).

Function Notation: Evaluating and Interpreting f(x)

1

f(x) = 2x² − 3x. Find f(−2).

2

f(x) = 5 − x². Solve f(x) = −4.

3

h(t) = −5t² + 15t + 1 is a height in metres. Find h(2) and say what it means.

Domain and Range of Linear and Quadratic Functions

1

State the range of f(x) = −3(x + 2)² + 7.

2

State the domain and range of parabola P.

3

y = x² is drawn only for −3 ≤ x ≤ 1. State the range.

Domain and Range in Real-World Contexts

1

h(t) = −5t² + 10t + 40 is a ball's height in metres. State the domain and range in context.

2

A(w) = w(18 − w) is the area of a rectangle. State the domain and range.

3

Is "the number of tickets sold" discrete or continuous?

Investigating Transformations: The Roles of a, h and k

1

Describe the transformations in g(x) = −2(x − 3)² + 1.

2

Write the equation of f(x) = x² compressed vertically by a factor of 1/3 and translated 4 left and 2 down.

3

Find the image of (2, 4) on y = x² under g(x) = 0.5(x − 1)² + 3.

Sketching Quadratic Functions by Transforming f(x) = x²

1

List the vertex and four more points to sketch g(x) = 2(x + 1)² − 3.

2

What is the step pattern for a = −0.5?

3

Use a sketch of g(x) = (x − 2)² − 4 to find its zeros.

From Vertex Form to Standard Form

1

Write f(x) = −3(x + 2)² + 5 in standard form.

2

Write f(x) = 0.5(x − 4)² − 1 in standard form.

3

Is (x + 5)² equal to x² + 25? Explain.

From Standard Form to Vertex Form: Completing the Square

1

Write f(x) = x² − 8x + 3 in vertex form.

2

Write f(x) = 2x² + 12x + 7 in vertex form.

3

Write f(x) = −x² + 3x + 1 in vertex form and state the maximum value.

Just for fun

You have now met one function wearing three outfits, doing yoga stretches, flipping upside down, throwing snowballs, and running a ticket booth. Not bad for something that started life as plain old x².

Sketching from Factored Form: Using the Zeros to Find the Vertex

1

For f(x) = (x + 2)(x − 6), state the zeros, axis, vertex and y-intercept.

2

Find the maximum value of f(x) = −0.5(x − 1)(x − 9).

3

A parabola has zeros −3 and 5 and passes through (1, 8). Find its equation in factored form. (It is parabola P.)

Comparing the Three Forms of a Quadratic Function

1

Write parabola P in all three forms.

2

Which form answers "When does the ball land?" directly?

3

Does f(x) = −(x − 2)² − 3 have real zeros? Can it be written in factored form?

Sketching from Standard Form: Choosing a Strategy

1

Find the vertex and y-intercept of f(x) = x² − 6x + 2.

2

Use partial factoring to find the vertex of f(x) = 3x² + 12x + 5.

3

State the range of f(x) = −2x² + 8x + 1.

Collecting and Graphing Data That Can Be Modelled by a Quadratic

1

Give one primary and one secondary source of quadratic data.

2

Is y = 4, 7, 12, 19, 28 (x = 0 to 4) quadratic?

3

Why should the x-values be equally spaced when you use differences?

Finding a Quadratic Model for Data

1

A curve of best fit has estimated vertex (3, 12) and passes through (0, 3). Find its equation.

2

A curve of best fit has zeros 2 and 10 and passes through (6, −8). Find its equation.

3

A quadratic regression has R² = 0.99. What does that tell you?

Solving Problems: Heights, Motion and Arches

1

h(t) = −5t² + 20t + 1.5. Find the maximum height and when it occurs.

2

For the same ball, when is it 16.5 m high?

3

When does it hit the ground? Round to two decimal places.

Solving Problems: Area, Revenue and Profit

1

200 m of fence encloses a rectangle. What is the largest possible area?

2

A club's revenue is R(x) = (20 + x)(500 − 20x) dollars when the $20 price rises by x dollars. Find the best price and the maximum revenue.

3

P(x) = −(x − 5)(x − 25). Find the break-even points and the maximum.

Check your understanding

Quiz

1.

Which relation is NOT a function?

2.

f(x) = x² − 4x. What is f(−1)?

3.

What is the range of f(x) = 2(x − 1)² − 3?

4.

g(x) = (x + 3)² − 2 is f(x) = x² translated…

5.

What is (x − 1)² + 4 in standard form?

6.

What is x² + 10x + 20 in vertex form?

7.

What is the vertex of f(x) = (x − 2)(x + 8)?

8.

Which form shows the zeros directly?

9.

h(t) = −5t² + 30t is a height in metres. What is the maximum height?

10.

Measured data with equal x-steps has roughly constant second differences. The best model is probably…

For parents and teachers

How to use the results: each section of the review matches one lesson. If your teen makes a particular kind of mistake, revisit that lesson:

  • Thinking repeated outputs break the function rule; missing ± with y² → Relations and Functions: What Makes a Function?.
  • Reading f(x) as f times x; sign errors when substituting negatives → Function Notation: Evaluating and Interpreting f(x).
  • Mixing up domain and range; wrong inequality for parabolas that open down; missing the vertex in a partial graph → Domain and Range of Linear and Quadratic Functions.
  • Keeping negative times or widths; smooth curves for counted quantities → Domain and Range in Real-World Contexts.
  • Moving the wrong way for h; calling a stretch "wider" → Investigating Transformations: The Roles of a, h and k.
  • Steps from the origin instead of the vertex; forgetting to multiply steps by a → Sketching Quadratic Functions by Transforming f(x) = x².
  • (x + 3)² = x² + 9 → From Vertex Form to Standard Form.
  • Forgetting to multiply by a when a term leaves the bracket → From Standard Form to Vertex Form: Completing the Square.
  • Sign errors reading zeros; forgetting to substitute for the vertex's y → Sketching from Factored Form: Using the Zeros to Find the Vertex.
  • Not knowing which form to use; expecting every quadratic to factor → Comparing the Three Forms of a Quadratic Function.
  • Using h = b ÷ (2a) without the negative; partial-factoring errors → Sketching from Standard Form: Choosing a Strategy.
  • Joining dots; using differences with unequal steps → Collecting and Graphing Data That Can Be Modelled by a Quadratic.
  • Using a point too close to the vertex to find a; trusting extrapolation → Finding a Quadratic Model for Data.
  • Solving h(t) = 0 when another height is asked; giving one of two times → Solving Problems: Heights, Motion and Arches.
  • Answering with x when the question asks for the price or the dimensions → Solving Problems: Area, Revenue and Profit.

The companion unit Quadratic Expressions and Equations covers expanding, factoring, the quadratic formula and the discriminant, which this unit uses throughout. In Grade 12, these ideas carry on in courses such as Mathematics for College Technology (MCT4C).