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Amy Ferguson Moncure

Algebra 1 · Unit 10

Linear and Quadratic Features

Identify slope, intercepts, vertex, domain, and range for linear and quadratic functions, including in context.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Linear

Constant slope. y-intercept is the start. Domain on the plane is all reals unless a story cuts it.

Quadratic

Vertex is a max or min. Zeros are x-intercepts. Range is one side of the vertex.

Read the representation

From an equation, pick out m and b, or a, vertex, and c. From a graph, read intercepts and the turning point.

In context

Slope is a rate. A vertex is a highest or lowest value. A zero can be empty, paid off, or on the ground.

Level 1

Easy

Name a feature from y = mx + b or a simple parabola.

Example
For y = 3x − 6, name the slope and the y-intercept. For f(x) = x² + 1, name the vertex.

Linear: slope 3, y-intercept (0, −6).

Quadratic: vertex (0, 1), opens up.

Answer: slope 3; (0, −6); vertex (0, 1)

1
Find the zero of y = 2x + 8.
Show Answer

0 = 2x + 8

x = −4

Zero (−4, 0)

Answer: (−4, 0)

2
Does f(x) = −1x² + 4 open up or down? Is the vertex a max or a min?
Show Answer

Opens down. Maximum at (0, 4).

Answer: down; max

3
A cab is C = 3m + 5 dollars after m miles. What do 3 and 5 mean?
Show Answer

3 dollars per mile. 5 dollar start.

Answer: rate 3; start 5

4
f(x) = (x − 2)² + 3. Name domain and range on the plane.
Show Answer

Domain all reals.

Range y ≥ 3.

Answer: all reals; y ≥ 3

5
The graph of a line through (0, 4) and (2, 0). Name slope and the zero.
Show Answer

m = −42 = −2

Zero (2, 0)

Answer: slope −2; (2, 0)

Level 2

Medium

Compare a linear feature to a quadratic feature in a story.

Example
A tank V = 40 − 5t gallons. A jump’s height h(t) = −16t² + 32t. Interpret each intercept at t = 0, and say which model has a maximum.

V(0) = 40: the tank starts with 40 gallons. Linear, no max (it just drains).

h(0) = 0: the jump starts on the ground. Quadratic, opens down, so a maximum at the vertex.

Answer: start 40 gal; start on ground; height has a max

1
Find the vertex of f(x) = x² − 6x + 5 and the slope of y = −3x + 1.
Show Answer

Vertex x = 3, f(3) = −4, vertex (3, −4).

Slope −3.

Answer: (3, −4); slope −3

2
A profit P(n) = −2n² + 20n and a cost C(n) = 4n + 10. Which has a maximum, and at what n?
Show Answer

P is quadratic opening down. n = 5, P(5) = 50.

C is linear, no max.

Answer: P at n = 5

3
Domain in the story: tickets T = 12n, n = 0, 1, 2, … versus h(t) = −16t² + 32t, t ≥ 0 until landing. Contrast.
Show Answer

T: whole numbers of tickets.

h: times from 0 to the positive zero, any time in between.

Answer: discrete vs a time interval

4
Zeros of y = (x − 1)(x − 5) and of y = 2x − 8. What does each zero mean if y is height in a story?
Show Answer

Quadratic: two times at height 0 (start and land, or two roots).

Linear: one time when height is 0.

Answer: two zeros vs one

5
Range of y = 4 and of y = x² + 4 on the plane.
Show Answer

Horizontal line: range only 4.

Parabola: y ≥ 4.

Answer: only 4 vs y ≥ 4

Level 3

Hard

A restricted quadratic and a linear rate in the same setting.

Example
A ball h(t) = −16t² + 48t + 4, 0 ≤ t ≤ 3, and a platform height L(t) = 4 (constant). Compare domain, range, and intercepts in the story.

Both start at 4 when t = 0.

h has vertex t = 32, h = 40. Range of h: 4 ≤ h ≤ 40 on this interval (h(3) = 4).

L has range only 4. Domain both 0 ≤ t ≤ 3 if they share the window.

Answer: h ranges 4 to 40; L stays 4

1
f(x) = 2x + 1 and g(x) = x² − 4. Find all features asked: f’s slope, g’s zeros, both y-intercepts.
Show Answer

f slope 2, f(0) = 1.

g zeros ±2, g(0) = −4.

Answer: slope 2; zeros ±2; intercepts 1 and −4

2
A line through (0, 8) with slope −2 empties a tank. A parabola h(t) = −16t² + 32t + 8 starts at the same 8. Which hits 0 first if t is seconds?
Show Answer

Line: 8 − 2t = 0, t = 4.

h(t) = 0: −16t² + 32t + 8 = 0, 2t² − 4t − 1 = 0, t = 4 ± √244 = 2 ± √62.

The positive root is about 2.2 s, before t = 4.

The jump hits the ground first.

Answer: the parabola, about 2.2 s

3
Why is the range of a non-horizontal line all reals, but the range of x² is not?
Show Answer

A non-horizontal line takes every y. x² never goes below 0.

Answer: line all y; x² y ≥ 0

4
Vertex of −1(x − 3)² + 10 and intercept of y = 10. Same maximum height in a story?
Show Answer

Both reach 10. The parabola reaches 10 only at x = 3. The line is always 10.

Answer: same 10, different when

5
A linear price p = 20 − n and a quadratic revenue R = n(20 − n). Interpret the vertex of R using the line p.
Show Answer

R = 20n − n², vertex n = 10, R = 100.

When 10 items sell at 10 dollars each (the line p = 10 at n = 10).

Answer: n = 10, p = 10, R = 100

Super Challenge

Super Challenge

1
A linear model and a quadratic model both pass through (0, 5) and (4, 13). Give one of each, and explain which extra point would decide the family.
Show Answer

Line: slope 2, y = 2x + 5.

One quadratic: many exist. Example y = x² − 2x + 5 also hits (0, 5) and (4, 13).

A third point, say x = 2: line gives 9, that quadratic gives 5. The data at x = 2 choose.

Answer: need a third point

2
List one feature each model always has that the other does not: y = mx + b (m ≠ 0) versus y = ax² + bx + c (a ≠ 0).
Show Answer

Linear: constant slope, no vertex.

Quadratic: a vertex, changing slope.

Answer: slope vs vertex

3
In a 0 ≤ t ≤ 5 story, f(t) = 2t + 1 and g(t) = −1(t − 2)² + 9. Which range is larger, and why does the window matter?
Show Answer

f goes from 1 to 11.

g: vertex (2, 9) in the window, g(0) = 5, g(5) = 0, so 0 ≤ g ≤ 9.

f’s range is longer on this window. Outside the window g would go down forever.

Answer: f on this window

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