# Linear and Quadratic Functions

**Section summaries**

Section 4.1 Linear Functions and Their Properties

A** linear function** is one of the form

f(x) = mx + b ,

where m gives the slope of its graph, and b gives the y-intercept of its graph.
The slope m

measures the rate of growth of the function, so a linear function is increasing
if m > 0 and

decreasing if m < 0.

Review problems: p284 #17,21,25,37,43,49

Section 4.2 Building Linear Functions from Data

In this section linear functions are constructed from data presented in various
ways.

Review problems: p290 #3,5,7,15,19,21

Section 4.3 Quadratic Functions and Their Properties

The **general form of a quadratic function** is

where (h, k) is the vertex of the graph (which is a parabola). You can see from
the formula

that h gives the left/right shift while k gives the up/down shift. The
coefficient a represents

a vertical stretch or compression. Since the basic member of this family is f(x)
= x^{2}, whose

graph opens up, the graph of
will open up if a is positive, and down

if a is negative. If the graph opens up, its height is minimum at the vertex, if
the graph

opens down, its height is maximum at the vertex.

If a quadratic is given in the form
then the x-coordinate of its

vertex is Since you already know the quadratic formula, you can remember it as

part of the formula:

This way of looking at the quadratic formula shows that if the graph has
x-intercepts, they

occur as points on either side of the line which is the
**axis of symmetry** of the

graph. The summary on page 301 explains the steps in graphing a quadratic
function.

Review problems: p302 #13,15,27,31,37,43,55,61,81,83

4.1 #29. Solve f(x) ≤g(x) for f(x) = 4x - 1 and g(x) = -2x + 5

(e) None of these

4.1 #37c. The cost (in dollars) of renting a truck is C(x)
= 0.25x+35, where x is the number

of miles driven. If you want the cost to be no more than $100, what is the
maximum number

of miles you can drive?

(a) 60

(b) 260

(c) 400

(d) 540

(e) None of these

4.3 A. If f(x) is a quadratic function whose graph has the
vertex (h, k), which one is the

correct form of the function?

4.3 #42 If f(x) = x^{2} - 2x - 3, then the vertex of the
graph of f(x) is

(e) None of these

4.3 B. Find the vertex of the quadratic function f(x) =
2x^{2} - 4x + 9.

(e) None of these

4.3 C. Find the axis of symmetry of the graph of f(x) = 4x^{2}
- 8x + 3.

(e) None of these

4.3 D. Let f(x) = 4x^{2} - 8x + 3. Find the x and
y-intercepts, if any.

(e) None of these

4.3 #55. Find the equation of the quadratic function whose
graph has vertex (-3, 5) and

y-intercept -4.

(e) None of these

4.3 #61. Find the minimum value of the function f(x) = 2x^{2}
+ 12x - 3.

(e) None of these

4.3 #62. Find the minimum value of the quadratic function
f(x) = 4x^{2}
- 8x + 3.

4.3 #81. Suppose that the manufacturer of a gas clothes
dryer has found that when the

unit price is p dollars the revenue R (in dollars) is R(p) = -4p^{2}+4000p. What
is the largest

possible revenue? That is, find the maximum value of the revenue function.

(e) None of these

4.3 E. A store selling calculators has found that, when
the calculators are sold at a price

of p dollars per unit, the revenue R (in dollars) as a function of the price p
is R(p) =

-750p^{2} + 15000p. What is the largest possible revenue? That is, find the maximum
value

of the revenue function.

(e) None of these

**Answer Key**

4.1 #29. (d)

4.1 #37c. (b)

4.3 A. (b)

4.3 #42 (c)

4.3 B. (d)

4.3 C. (a)

4.3 D. (b)

4.3 #55. (c)

4.3 #61. (c)

4.3 #62. (b)

4.3 #81. (b)

4.3 E. (d)

**Solutions**

4.3 A. If f(x) is a quadratic function whose graph has the vertex (h, k), which
one is the

correct form of the function?

Solution: (b) f(x) = a(x - h)^{2} + k

4.3 B. Find the vertex of the quadratic function f(x) = 2x^{2} - 4x + 9.

Solution: (d) The text gives this formula: the x-coordinate of the vertex of the
graph

of f(x) = ax^{2} +bx+c is
In this example, a = 2 and b = -4, so the vertex occurs

at
Then f(1) = 2 - 4 + 9 = 7 gives the y-coordinate.

If you forget the formula, you can always complete the square:

so h = 1 and k = 7 and the vertex is (1, 7).

4.3 C. Find the axis of symmetry of the graph of f(x) = 4x^{2} - 8x + 3.

Solution: (a) The axis of symmetry passes through the vertex, which has
x-coordinate

The axis of symmetry is the line x = 1.

Again, if you forget the formula, complete the square:

This shows that the vertex is at (1,-1).

4.3 D. Let f(x) = 4x^{2} - 8x + 3. Find the x and y-intercepts, if any.

Solution: (b) Since f(0) = 3, the y-intercept is (0, 3).

To find the x-intercept, solve 4x^{2} - 8x + 3 = 0. This can be factored as

(2x - 1)(2x - 3), so 2x - 1 = 0 or 2x - 3 = 0, giving the x-intercepts
and

4.3 #62. Find the minimum value of the quadratic function
f(x) = 4x^{2} - 8x + 3.

Solution: (b) The minimum value occurs at the vertex, which has x-coordinate

Then f(1) = -1 is the minimum height.

4.3 E. A store selling calculators has found that, when the calculators are sold
at a price

of p dollars per unit, the revenue R (in dollars) as a function of the price p
is R(p) =

-750p^{2} + 15000p. What is the largest possible revenue? That is, find the maximum
value

of the revenue function.

Solution: (d) The graph is a parabola, opening down, so to find the maximum value
we

need to find the y-coordinate of the vertex. We get

The maximum revenue is R(10) = -750(10)^{2} + 15000(10) = -75000 + 150000 =
75000.