### CHAPTER 1:

### Graphs, Functions,

### and Models

**1.1 Introduction to Graphing**

1.2 Functions and Graphs

1.3 Linear Functions, Slope, and Applications 1.4 Equations of Lines and Modeling

### 1.1

### Introduction to Graphing

_{ Plot points.}

_{ Determine whether an ordered pair is a solution of an }

equation.

_{ Find the }_{x}_{-and }_{y}_{-intercepts of an equation of the form }

*Ax* + *By = C*.

_{ Graph equations.}

_{ Find the distance between two points in the plane and find the }

midpoint of a segment.

_{ Find an equation of a circle with a given center and radius, and }

given an equation of a circle in standard form, find the center and the radius.

To graph or **plot** a point, the first coordinate tells us to
move left or right from the origin. The second

coordinate tells us to move up or down.

Plot (3, 5).

Move 3 units left.

Next, we move 5 units up. Plot the point.

### Solutions of Equations

Equations in two variables have solutions (*x*, *y*) that are
ordered pairs.

**Example:** 2*x* + 3*y* = 18

### Examples

**a.**

Determine whether the
ordered pair (5, 7) is a
solution of 2*x* + 3*y* = 18.

2(5) + 3(7) ? 18

10 + 21 ? 18 11 = 18 FALSE

(5, 7) is not a solution.

**b.**

Determine whether the
ordered pair (3, 4) is a
solution of 2*x* + 3*y* = 18.

2(3) + 3(4) ? 18 6 + 12 ? 18 18 = 18 TRUE

### Graphs of Equations

To **graph an equation **is to make a drawing that

*x*

### -Intercept

The point at which the graph crosses the x-axis.

An x**-intercept** is a point (a, 0). To find a, let y = 0 and
solve for x.

Example: Find the x-intercept of 2x + 3y = 18. 2x + 3(0) = 18

2x = 18 x = 9

*y*

### -Intercept

The point at which the graph crosses the *y*-axis.

A **y****-intercept** is a point (0, *b*). To find *b*, let *x* = 0 and

solve for *y*.

Example: Find the *y*-intercept of 2*x* + 3*y* = 18.
2(0) + 3*y* = 18

3*y* = 18
*y* = 6

### Example

We already found the *x*-intercept: **(9, 0)**

We already found the *y*-intercept: **(0, 6)**

We find a third solution as a check. If *x* is replaced
with 5, then

Graph 2*x* + 3*y* = 18.

2 _{5} _{3}*y* _{18}

10 _{3}*y* _{18}

3*y* _{8}

*y* 8

3

Thus, is a solution.5, 8

### Example

(continued) Graph: 2*x* + 3*y* = 18.

*x*-intercept:

**(9, 0)**

*y*-intercept:

**(0, 6)**

Third point:

5, 8 3

### Example

Graph *y = x*2 – 9*x* – 12 .

(12, 24)
24
12
–2
32
32
26
12
–2
24
*y*
(10, –2)
10

(5, 32) 5

(4, 32) 4

(2, 26) 2

(0, 12) 0

(1, –2)

1

(3, 24)

3

(*x*, *y*)

*x*

### The Pythagorean Theorem

*a*

*b*

*c*

*a*

2### +

*b*

2### =

*c*

2
*Which side is *

*the hypotenuse?*

*a*

2### =

*c*

2 ### –

*b*

2
The right angle points to the hypotenuse.

### 6

### 8

*c*

### Calculate side c.

*c*

2### =

### 8

2### + 6

2*c*

2### = 64

### + 36

*c*

2### = 100

### 100

*c*

###

*c*

### = 10

When calculating the

hypotenuse, we add the area of the squares of the other two sides.

### Tanya is making a party hat using a

### cone made out of paper. Determine

### the height of the cone.

*b*

2### =

*c*

2### –

*a*

2
*h*

2### = 144

* h *

### = 12 cm

*h*

2### = 13

2### – 5

2*h*

2### = 169– 25

### 144

###

*h*

h

5 cm

### The Distance Formula

The **distance *** d* between any two points

(*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) is given by

*d* _{(}*x*_{2} *x*_{1})2 _{} _{(}_{y}

### Example

Find the distance between (4,8) and (1,12)

2 2

2 1 2 1

### distance

###

### (

*x*

###

*x*

### )

###

### (

*y*

###

*y*

### )

### (4, 8)

### (1, 12)

2 2

### distance

###

### (1 4)

###

###

### (12 8)

###

2 2

### distance

###

### ( 3)

###

###

### (4)

### Example

*d* _{(}_{3}

###

_{2}

_{)}2

_{}

_{(}

_{}

_{6}

_{}

_{2}

_{)}2

*d* _{5}2 _{} _{(}_{}_{8)}2 _{} _{25} _{} _{64}

*d* _{89} _{9.4}

### Midpoint Formula

If the endpoints of a segment are (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}),
then the coordinates of the **midpoint** are

*x*_{1} *x*_{2}

2 ,

*y*_{1} *y*_{2}

2

### Example

Find the midpoint of a segment whose endpoints are (4, 2) and (2, 5).

_{4} _{2}

2 ,

_{2} _{5}

2

2

2 , 32

_{1, 3}

### Circles

A **circle** is the set of all points in a plane that are a

fixed distance *r* from a *center* (*h*, *k*).

The **equation of a circle **with center (*h, k*) and radius

*r*, in standard form, is

### Example

Find an equation of a circle having radius 5 and center (3, 7).

Using the standard form, we have
(*x* *h*)2 + (*y* *k*)2 = *r*2

[*x* 3]2 + [*y* (7)]2 = 52

### Ex. 2: Writing a Standard Equation of a

### Circle

The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle.

2 1 2

2 1

2 ) ( )

(*x* *x* *y* *y*

r =

r = _{(}_{5} _{1}_{)}2 _{(}_{1} _{2}_{)}2
r = _{(}_{4}_{)}2 _{} _{(}_{}_{3}_{)}2

r = _{16} _{9}

r = _{25}

Use the Distance Formula Substitute values.

Simplify.

Simplify.

Addition Property

### Ex. 2 (Cont’d): Writing a Standard

### Equation of a Circle

The point (1, 2) is on a circle whose center is (5, -1). Write a standard equation of the circle.

(x – h)2 + (y – k)2 = r2 Standard equation of a circle.

[(x – 5)]2 + [y –(-1)]2 = 52 Substitute values.