write a function rule for the statement. then graph the function.
1. the output is ten less than the input. 2. the output is one-third of the input.
solve the system.
3. $y = x + 5$
$y = -3x + 1$
4. $x + y = -4$
$6x + 2y = 4$
5. $-4x + 3y = 14$
$y = 2x + 8$

concepts, skills, & problem solving
writing and graphing functions the table shows a familiar pattern from geometry. (a) determine what the variables $x$ and $y$ represent. then write a function rule that relates $y$ to $x$. (b) is the function a linear function? explain your reasoning. (see exploration 1, p. 289.)
6.
|$x$|1|2|3|4|5|
|----|----|----|----|----|----|
|$y$|2|4|6|8|10|
7.
|$x$|1|2|3|4|5|
|----|----|----|----|----|----|
|$y$|$\pi$|$2\pi$|$3\pi$|$4\pi$|$5\pi$|

writing linear functions use the graph or table to write a linear function that relates $y$ to $x$.
8.
9.
10.
|$x$|-8|-4|0|4|
|----|----|----|----|----|
|$y$|2|1|0|-1|
11.
|$x$|-3|0|3|6|
|----|----|----|----|----|
|$y$|3|5|7|9|
12. interpreting a linear function the table shows the length $y$ (in inches) of a persons hair after $x$ months.
a. write and graph a linear function that relates $y$ to $x$
b. interpret the slope and the $y$-intercept
|months, $x$|hair length, $y$|
|----|----|
|0|11.0|
|1|12.5|
|2|14.0|

section 7.3 linear functions 293

Question

write a function rule for the statement. then graph the function.
1. the output is ten less than the input. 2. the output is one-third of the input.
solve the system.
3. $y = x + 5$
   $y = -3x + 1$
4. $x + y = -4$
   $6x + 2y = 4$
5. $-4x + 3y = 14$
   $y = 2x + 8$

concepts, skills, & problem solving
writing and graphing functions the table shows a familiar pattern from geometry. (a) determine what the variables $x$ and $y$ represent. then write a function rule that relates $y$ to $x$. (b) is the function a linear function? explain your reasoning. (see exploration 1, p. 289.)
6.
|$x$|1|2|3|4|5|
|----|----|----|----|----|----|
|$y$|2|4|6|8|10|
7.
|$x$|1|2|3|4|5|
|----|----|----|----|----|----|
|$y$|$\pi$|$2\pi$|$3\pi$|$4\pi$|$5\pi$|

writing linear functions use the graph or table to write a linear function that relates $y$ to $x$.
8.
9.
10.
|$x$|-8|-4|0|4|
|----|----|----|----|----|
|$y$|2|1|0|-1|
11.
|$x$|-3|0|3|6|
|----|----|----|----|----|
|$y$|3|5|7|9|
12. interpreting a linear function the table shows the length $y$ (in inches) of a persons hair after $x$ months.
a. write and graph a linear function that relates $y$ to $x$
b. interpret the slope and the $y$-intercept
|months, $x$|hair length, $y$|
|----|----|
|0|11.0|
|1|12.5|
|2|14.0|

section 7.3 linear functions 293

Accepted Answer

### Problems 1-2: Write function rules
# Explanation:
## Step1: Define input/output
Let input = $x$, output = $y$.
## Step2: Translate statement 1
Output = input - 10
$y = x - 10$
## Step3: Translate statement 2
Output = $\frac{1}{3}$ of input
$y = \frac{1}{3}x$
# Answer:
1. $y = x - 10$ (graph is a line with slope 1, y-intercept -10)
2. $y = \frac{1}{3}x$ (graph is a line with slope $\frac{1}{3}$, y-intercept 0)

---

### Problems 3-5: Solve systems of equations
# Explanation:
#### Problem 3
## Step1: Substitute $y=x+5$
Substitute into $y=-3x+1$:
$x + 5 = -3x + 1$
## Step2: Solve for $x$
$x + 3x = 1 - 5$
$4x = -4$
$x = -1$
## Step3: Find $y$
$y = -1 + 5 = 4$

#### Problem 4
## Step1: Isolate $y$ from first eq
$y = -4 - x$
## Step2: Substitute into second eq
$6x + 2(-4 - x) = 4$
$6x - 8 - 2x = 4$
## Step3: Solve for $x$
$4x = 12$
$x = 3$
## Step4: Find $y$
$y = -4 - 3 = -7$

#### Problem 5
## Step1: Substitute $y=2x+8$
Substitute into $-4x + 3y = 14$:
$-4x + 3(2x + 8) = 14$
## Step2: Simplify and solve $x$
$-4x + 6x + 24 = 14$
$2x = 14 - 24$
$2x = -10$
$x = -5$
## Step3: Find $y$
$y = 2(-5) + 8 = -2$
# Answer:
3. $(-1, 4)$
4. $(3, -7)$
5. $(-5, -2)$

---

### Problem 6: Geometry function rule
# Explanation:
## Step1: Identify variables
$x$ = base of right triangle, $y$ = height (or vice versa; pattern matches $y=2x$)
## Step2: Write function rule
$\frac{y}{x} = 2$, so $y = 2x$
## Step3: Check linearity
Function fits $y=mx+b$ ($m=2, b=0$), so linear.
# Answer:
(a) $x$ = base of the right triangle, $y$ = height of the right triangle; function rule: $y=2x$
(b) Yes, it is linear. It follows the form $y=mx+b$ where $m=2$ and $b=0$, with a constant rate of change.

---

### Problem 7: Geometry function rule
# Explanation:
## Step1: Identify variables
$x$ = diameter of the circle, $y$ = circumference of the circle
## Step2: Write function rule
$\frac{y}{x} = \pi$, so $y = \pi x$
## Step3: Check linearity
Function fits $y=mx+b$ ($m=\pi, b=0$), so linear.
# Answer:
(a) $x$ = diameter of the circle, $y$ = circumference of the circle; function rule: $y=\pi x$
(b) Yes, it is linear. It follows the form $y=mx+b$ where $m=\pi$ and $b=0$, with a constant rate of change.

---

### Problems 8-11: Write linear functions
# Explanation:
#### Problem 8
## Step1: Find slope $m$
Use points $(1,3)$ and $(2,5)$:
$m = \frac{5-3}{2-1} = 2$
## Step2: Find y-intercept $b$
$3 = 2(1) + b$ → $b=1$
## Step3: Write function
$y=2x+1$

#### Problem 9
## Step1: Identify slope and intercept
Horizontal line: slope $m=0$, y-intercept $b=3$
## Step2: Write function
$y=3$

#### Problem 10
## Step1: Find slope $m$
Use points $(0,0)$ and $(4,-1)$:
$m = \frac{-1-0}{4-0} = -\frac{1}{4}$
## Step2: Find y-intercept $b$
$b=0$ (from point $(0,0)$)
## Step3: Write function
$y=-\frac{1}{4}x$

#### Problem 11
## Step1: Find slope $m$
Use points $(0,5)$ and $(3,7)$:
$m = \frac{7-5}{3-0} = \frac{2}{3}$
## Step2: Find y-intercept $b$
$b=5$ (from point $(0,5)$)
## Step3: Write function
$y=\frac{2}{3}x+5$
# Answer:
8. $y=2x+1$
9. $y=3$
10. $y=-\frac{1}{4}x$
11. $y=\frac{2}{3}x+5$

---

### Problem 12: Interpret linear function
# Explanation:
## Step1: Find slope $m$
Use points $(0,11.0)$ and $(2,12.5)$:
$m = \frac{12.5-11.0}{2-0} = 0.75$
## Step2: Find y-intercept $b$
$b=11.0$ (from $x=0$)
## Step3: Write function
$y=0.75x+11.0$
## Step4: Interpret slope/intercept
Slope = 0.75: hair grows 0.75 inches per month.
Y-intercept = 11.0: initial hair length is 11.0 inches at 0 months.
# Answer:
a. Linear function: $y=0.75x+11.0$ (graph is a line with slope 0.75, y-intercept 11.0)
b. Slope: The person's hair grows 0.75 inches each month. Y-intercept: The person's hair length was 11.0 inches at the start (0 months).

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problems 1-2: Write function rules

Step1: Define input/output

Step2: Translate statement 1

Step3: Translate statement 2

Problem 3

Step1: Substitute $y=x+5$

Step2: Solve for $x$

Step3: Find $y$

Problem 4

Step1: Isolate $y$ from first eq

Step2: Substitute into second eq

Step3: Solve for $x$

Step4: Find $y$

Problem 5

Step1: Substitute $y=2x+8$

Step2: Simplify and solve $x$

Step3: Find $y$

Step1: Identify variables

Step2: Write function rule

Step3: Check linearity

Answer:

Problems 3-5: Solve systems of equations