Chapter 1 Review / Test Prep — Cornerstone MathLab

Step 1 of 4

Quick Check

Ten quick questions across the whole chapter.

How this review works

Practise → diagnose → fix mistakes → test yourself → know what to review. Questions are mixed on purpose. Hints are available until Test Yourself.

True or false

{(−2, 5), (0, 5), (3, 5)}

This relation is a function.

TrueFalse

Multiple choice

x: 0, 1, 2, 3 y: 4, 7, 12, 19

Linear, quadratic or neither?

aLinearbQuadraticcNeither

Numerical answer

f(x) = x² − 3x. Find f(−2).

Your answer

Multiple choice

For g(x) = 4(x − 5)² − 7, what are a, h and k?

aa = 4, h = 5, k = −7ba = 4, h = −5, k = −7ca = 4, h = 5, k = 7da = −4, h = 5, k = −7

Multiple choice

What is the vertex of y = −(x + 6)² + 2?

a(6, 2)b(−6, 2)c(−6, −2)d(2, −6)

Multiple choice

Compared with y = x², the graph of y = −0.5(x − 1)² + 8…

aopens up and is narrowerbopens down and is narrowercopens down and is widerdopens up and is wider

Numerical answer

The axis of symmetry of y = 3(x + 4)² − 1 is x = c. Enter c.

Your answer

Multiple choice

What is the range of y = 2(x − 3)² − 5?

a{y ∈ ℝ | y ≥ −5}b{y ∈ ℝ | y ≤ −5}c{y ∈ ℝ | y ≥ 3}d{y ∈ ℝ}

Multiple choice

y = x² is transformed into y = x² − 6. Which transformation occurred?

aTranslation 6 units rightbTranslation 6 units leftcTranslation 6 units downdVertical compression by 6

Multiple choice

Mapping diagram: Locker 12 → Ana, Locker 15 → Ben, Locker 18 → Ana, Locker 12 → Chen

Which statement is correct?

aFunction — Ana is allowed to appear twice.bNot a function — Locker 12 is paired with two people.cNot a function — Ana has two lockers.dFunction — every locker has a person.

Step 2 of 4

Mix It Up

Decide which mathematics each problem needs.

Problem 1 — The marble ramp

One table, five questions. Work through them in order.

Multiple choice

A marble rolls down a ramp. Distance d (m) after t seconds: t: 0, 1, 2, 3, 4 d: 0, 1.5, 6, 13.5, 24

A. Is the relation a function?

aYes — each time has exactly one distance.bNo — the distances are not equally spaced.cNo — distance increases.dYes — because 0 is included.

Multiple choice

B. What are the first differences of the marble table?

a1.5, 4.5, 7.5, 10.5b1.5, 6, 13.5, 24c3, 3, 3d1.5, 3, 4.5, 6

Numerical answer

C. What is the (constant) second difference?

Your answer

Multiple choice

D. Classify the marble relationship.

aLinear — degree 1bQuadratic — degree 2cNeitherdLinear, because the time steps are equal

Multiple choice

E/F. Which uses function notation correctly AND interprets it in context?

ad(13.5) = 3: after 13.5 s the marble has rolled 3 m.bd(3) = 13.5: 3 seconds after release, the marble has rolled 13.5 m.cd × 3 = 13.5: the marble rolls 4.5 m per second.dd(3) = 13.5: the marble rolls 13.5 m during the 3rd second only.

Problem 2 — Analyse g(x) = −2(x + 3)² + 5

Connect the equation to the graph's features.

Multiple choice

g(x) = −2(x + 3)² + 5

A–C. Which gives a, h, k, the vertex and the axis of symmetry?

aa = −2, h = −3, k = 5; vertex (−3, 5); axis x = −3ba = −2, h = 3, k = 5; vertex (3, 5); axis x = 3ca = 2, h = −3, k = 5; vertex (−3, 5); axis y = 5da = −2, h = −3, k = 5; vertex (5, −3); axis x = 5

Multiple choice

D/E. For g(x) = −2(x + 3)² + 5, which description is complete and correct?

aOpens down; reflection in the x-axis, vertical stretch by a factor of 2, translation 3 left and 5 up.bOpens down; vertical compression by a factor of 2, translation 3 right and 5 up.cOpens up; vertical stretch by 2, translation 3 left and 5 up.dOpens down; reflection in the y-axis, stretch by 2, translation 3 left and 5 up.

Multiple choice

F/G. State the domain and range of g(x) = −2(x + 3)² + 5.

aD: {x ∈ ℝ}, R: {y ∈ ℝ | y ≤ 5}bD: {x ∈ ℝ}, R: {y ∈ ℝ | y ≥ 5}cD: {x ∈ ℝ | x ≤ −3}, R: {y ∈ ℝ | y ≤ 5}dD: {x ∈ ℝ | x ≥ −3}, R: {y ∈ ℝ}

Problem 3 — Read the graph

Use only the graph to answer.

Mystery parabola

Multiple choice

Use the graph shown above.

What are the vertex and axis of symmetry?

a(2, −4); x = 2b(−4, 2); x = −4c(2, −4); y = −4d(−2, −4); x = −2

Multiple choice

Use the graph shown above.

Which gives the opening, domain and range?

aUp; D {x ∈ ℝ}; R {y ∈ ℝ | y ≥ −4}bUp; D {x ∈ ℝ | x ≥ 2}; R {y ∈ ℝ | y ≥ −4}cDown; D {x ∈ ℝ}; R {y ∈ ℝ | y ≤ −4}dUp; D {x ∈ ℝ}; R {y ∈ ℝ | y ≥ 2}

Multiple choice

Use the graph shown above. It passes through (0, −2) and (4, −2).

Which equation is most likely?

ay = 0.5(x − 2)² − 4by = 2(x − 2)² − 4cy = 0.5(x + 2)² − 4dy = −0.5(x − 2)² − 4

Fix the Graph

Diagnose another student's graph, then correct it.

Fix the Graph

Each round shows an equation, a student's incorrect graph and their reasoning. Name what went wrong, then use the controls to correct the graph.

Fix the Graph — Round 1 of 4 · Horizontal shift

Equation: g(x) = (x − 3)² + 1

Jordan's graph

Jordan's reasoning
“The bracket says − 3, so I moved the vertex 3 units left to (−3, 1).”

Step 1 · What went wrong?

Step 4 of 4

Test Yourself

Test conditions — no hints, results after you submit.

Test conditions

12 questions, no hints. Answer every question, then submit to see your results and your Test Readiness report.

Test Yourself

12 questions · no hints during the check. Answer them all, then check your score — you can retry as many times as you like.

Question 1 of 12
Which relation is NOT a function?
x: 2, 4, 2, 6 y: 1, 3, 5, 7{(1, 4), (2, 4), (3, 4)}y = x² − 9An upward parabola with vertex (0, −3)
Question 2 of 12
{(−3, 2), (0, −1), (4, 2), (5, 6)}
What are the domain and range?
D {−3, 0, 4, 5}; R {−1, 2, 6}D {−1, 2, 6}; R {−3, 0, 4, 5}D {−3, 0, 4, 5}; R {2, −1, 2, 6}D {x ∈ ℝ}; R {y ∈ ℝ}
Question 3 of 12
A drone descends to land. Height (m): t: 0, 1, 2, 3, 4 h: 50, 45, 36, 23, 6
Which model fits, and why?
Linear — the height decreases every second.Quadratic — first differences −5, −9, −13, −17; second differences constant at −4.Neither — first differences are not constant.Quadratic — first differences constant at −5.
Question 4 of 12
x: 0, 2, 3, 5 y: 1, 5, 7, 11. Dara says: “First differences are 4, 2, 4 — not constant — so it is not linear.”
Which explanation is mathematically strongest?
Dara is correct; the differences change.Dara is wrong: the x-values aren't equally spaced. Change per unit of x is 4/2, 2/1, 4/2 = 2 each time, so it is linear.Dara is wrong because the second differences are −2 and 2.Dara is wrong because every table with increasing y is linear.
Question 5 of 12
f(x) = 2x² − 3x + 1. What is f(−2)?
−1315−13
Question 6 of 12
T(m) is an oven's temperature (°C) m minutes after it is switched on.
Which statement best explains the difference between T(10) and T(m) = 180?
T(10) is the temperature after 10 minutes; T(m) = 180 asks when the oven reaches 180 °C.T(10) asks when the oven reaches 10 °C; T(m) = 180 is the temperature after 180 minutes.Both mean T × 10 = 180.T(10) is 10 times the temperature; T(m) = 180 is the final temperature.
Question 7 of 12
A student graphs y = 2(x + 4)² − 1 with vertex (4, −1). Which statement best explains the error?
The k-value was misread; the vertex is (4, 1).x + 4 means h = −4, so the vertex is (−4, −1); the student shifted right instead of left.The stretch factor 2 moves the vertex to (8, −1).There is no error; + 4 means move right 4.
Question 8 of 12
Compared with y = x², which describes y = −0.25(x − 2)²?
Reflection in the x-axis, vertical compression by 0.25, translation 2 rightReflection in the x-axis, vertical stretch by 4, translation 2 rightVertical compression by 0.25, translation 2 leftReflection in the y-axis, vertical compression by 0.25, translation 2 down
Question 9 of 12
The parent point (2, 4) on y = x² is transformed by y = −2(x − 1)² + 3. Where does it land?
(3, −5)(1, −5)(3, 11)(3, −1)
Question 10 of 12
Which explanation of how to graph y = −3(x + 2)² + 5 from y = x² is strongest?
Translate 2 left, then stretch by 3 and reflect in the x-axis, then translate 5 up — the vertical shift comes last so it isn't stretched or reflected.Translate 5 up, then stretch by 3 and reflect, then translate 2 left — order never matters.Translate 2 right, stretch by 3, reflect, translate 5 up.Reflect in the y-axis, compress by 3, translate 2 left and 5 up.
Question 11 of 12
A soccer ball is kicked from the ground. Its height is h(t) = −5(t − 3)² + 45 metres; it lands at t = 6 s.
What are the contextual domain and range?
D {t ∈ ℝ | 0 ≤ t ≤ 6}; R {h ∈ ℝ | 0 ≤ h ≤ 45}D {t ∈ ℝ}; R {h ∈ ℝ | h ≤ 45}D {t ∈ ℝ | 0 ≤ t ≤ 45}; R {h ∈ ℝ | 0 ≤ h ≤ 6}D {t ∈ ℝ | 0 ≤ t ≤ 3}; R {h ∈ ℝ | h ≥ 45}
Question 12 of 12
Lee says the range of y = −4(x − 1)² + 7 is {y ∈ ℝ | y ≥ 7}. Which response is strongest?
Lee is correct because k = 7.Lee reversed the inequality: a = −4 < 0, so the parabola opens down and 7 is a maximum. Range: y ≤ 7.The range is y ≤ 1 because h = 1.The range is all real numbers because parabolas go on forever.

Answer every question to check your work.

Before your test

1. Check whether x-values are equally spaced before using finite differences.

2. For vertex form, identify a, h and k carefully.

3. Remember the sign inside (x − h): (x + 4) means h = −4.

4. Use the vertex and the opening direction to determine the range.

5. Use brackets when substituting negative inputs: (−3)² = 9.

6. When stuck, connect the equation, the table and the graph.

Problem 1 — The marble ramp

Problem 2 — Analyse g(x) = −2(x + 3)² + 5

Problem 3 — Read the graph

More mixed problems

Test Yourself

Before your test