given f(x)=3x + 5, describe how the graph of g compares with the graph of f. see examples 1, 2, and 3
15. g(x)=(3x + 5)+8
16. g(x)=(3x + 5)-4
17. g(x)=3(x + 10)+5
18. g(x)=3(x - 1)+5
19. g(x)=3(0.1x)+5
20. g(x)=5(3x + 5)
21. g(x)=3(2x)+5
22. g(x)=8(3x + 5)
given f(x)=2x + 3, describe how the value of k affects the slope and y - intercept of the graph of g compared to the graph of f. see example 3
23. g(x)=3(2x + 3)
24. g(x)=2(0.5x)+3
25. g(x)=\frac{1}{6}(2x + 3)
26. g(x)=2(\frac{1}{8})x+3
27. g(x)=(2x + 3)-3
28. g(x)=2(x + 0.5)+3
find the value of k for each function g. then describe the graph of g as a transformation of the graph of f.
29. g(x)=(x + 1)+k, f(x)=x + 1
30. g(x)=-2(x + k)+1, f(x)=-2x + 1

Question

Accepted Answer

# Explanation:
## Step1: Recall function - transformation rules
For a linear function $y = f(x)$, if $g(x)=f(x)+c$ ($c$ is a constant), the graph of $g(x)$ is a vertical - shift of the graph of $f(x)$ by $c$ units. If $g(x)=f(x + d)$ ($d$ is a constant), the graph of $g(x)$ is a horizontal - shift of the graph of $f(x)$ by $d$ units. If $g(x)=a\cdot f(x)$ ($a$ is a non - zero constant), the graph of $g(x)$ is a vertical stretch or compression of the graph of $f(x)$ by a factor of $a$.

### 15.
Given $f(x)=3x + 5$ and $g(x)=(3x + 5)+8=f(x)+8$.
The graph of $g(x)$ is a vertical shift of the graph of $f(x)$ up 8 units.

### 16.
Given $f(x)=3x + 5$ and $g(x)=(3x + 5)-4=f(x)-4$.
The graph of $g(x)$ is a vertical shift of the graph of $f(x)$ down 4 units.

### 17.
Given $f(x)=3x + 5$ and $g(x)=3(x + 10)+5=f(x + 10)$.
The graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ left 10 units.

### 18.
Given $f(x)=3x + 5$ and $g(x)=3(x - 1)+5=f(x - 1)$.
The graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ right 1 unit.

### 19.
Given $f(x)=3x + 5$ and $g(x)=3(0.1x)+5$. Let $u = 0.1x$, then $g(x)$ is a horizontal stretch of $f(x)$ by a factor of $\frac{1}{0.1}=10$ units.

### 20.
Given $f(x)=3x + 5$ and $g(x)=5(3x + 5)=5f(x)$.
The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 5.

### 21.
Given $f(x)=3x + 5$ and $g(x)=3(2x)+5$. Let $u = 2x$, then $g(x)$ is a horizontal compression of $f(x)$ by a factor of $\frac{1}{2}$.

### 22.
Given $f(x)=3x + 5$ and $g(x)=8(3x + 5)=8f(x)$.
The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 8.

### 23.
Given $f(x)=2x + 3$ and $g(x)=3(2x + 3)=3f(x)$.
The slope of $f(x)$ is $m_1 = 2$ and the $y$ - intercept is $b_1=3$. For $g(x)$, the slope is $m_2 = 3\times2 = 6$ and the $y$ - intercept is $b_2=3\times3 = 9$. The slope is multiplied by 3 and the $y$ - intercept is multiplied by 3.

### 24.
Given $f(x)=2x + 3$ and $g(x)=2(0.5x)+3$. Let $u = 0.5x$, the slope of $f(x)$ is $m_1 = 2$ and for $g(x)$ the slope is $m_2=2\times0.5 = 1$, the $y$ - intercept $b_2 = 3$. The slope is multiplied by 0.5 and the $y$ - intercept remains the same.

### 25.
Given $f(x)=2x + 3$ and $g(x)=\frac{1}{6}(2x + 3)=\frac{1}{6}f(x)$.
The slope of $f(x)$ is $m_1 = 2$ and for $g(x)$ the slope is $m_2=\frac{1}{6}\times2=\frac{1}{3}$, the $y$ - intercept is $b_2=\frac{1}{6}\times3=\frac{1}{2}$. The slope and $y$ - intercept are multiplied by $\frac{1}{6}$.

### 26.
Given $f(x)=2x + 3$ and $g(x)=2(\frac{1}{8})x+3$. The slope of $f(x)$ is $m_1 = 2$ and for $g(x)$ the slope is $m_2=2\times\frac{1}{8}=\frac{1}{4}$, the $y$ - intercept $b_2 = 3$. The slope is multiplied by $\frac{1}{8}$ and the $y$ - intercept remains the same.

### 27.
Given $f(x)=2x + 3$ and $g(x)=(2x + 3)-3=f(x)-3$.
The slope of $f(x)$ and $g(x)$ is $m = 2$, and the $y$ - intercept of $f(x)$ is $b_1 = 3$ and for $g(x)$ is $b_2=3 - 3=0$. The slope remains the same and the $y$ - intercept is decreased by 3.

### 28.
Given $f(x)=2x + 3$ and $g(x)=2(x + 0.5)+3=f(x + 0.5)$.
The slope of $f(x)$ and $g(x)$ is $m = 2$, and the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ left 0.5 units.

### 29.
For $f(x)=x + 1$ and $g(x)=(x + 1)+k$.
The $y$ - intercept of $f(x)$ is 1. Let's use a point on $f(x)$ like $(0,1)$. For $g(x)$, when $x = 0$, $y=(0 + 1)+k=1 + k$. The graph of $g(x)$ is a vertical shift of the graph of $f(x)$ by $k$ units. From the graph, when $x = 0$, $f(0)=1$ and $g(0)=4$, so $k=3$. The graph of $g(x)$ is a vertical shift of the graph of $f(x)$ up 3 units.

### 30.
For $f(x)=-2x + 1$ and $g(x)=-2(x + k)+1$.
The slope of $f(x)$ and $g(x)$ is $-2$. Let's use the $y$ - intercept. The $y$ - intercept of $f(x)$ is 1. For $g(x)$, when $x = 0$, $y=-2k + 1$. From the graph, the $y$ - intercept of $g(x)$ is 5. So $-2k+1 = 5$, $-2k=4$, $k=-2$. The graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ right 2 units.

# Answer:
15. The graph of $g(x)$ is a vertical shift of the graph of $f(x)$ up 8 units.
16. The graph of $g(x)$ is a vertical shift of the graph of $f(x)$ down 4 units.
17. The graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ left 10 units.
18. The graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ right 1 unit.
19. The graph of $g(x)$ is a horizontal stretch of the graph of $f(x)$ by a factor of 10.
20. The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 5.
21. The graph of $g(x)$ is a horizontal compression of the graph of $f(x)$ by a factor of $\frac{1}{2}$.
22. The graph of $g(x)$ is a vertical stretch of the graph of $f(x)$ by a factor of 8.
23. The slope and $y$ - intercept of $f(x)$ are multiplied by 3.
24. The slope of $f(x)$ is multiplied by 0.5 and the $y$ - intercept remains the same.
25. The slope and $y$ - intercept of $f(x)$ are multiplied by $\frac{1}{6}$.
26. The slope of $f(x)$ is multiplied by $\frac{1}{8}$ and the $y$ - intercept remains the same.
27. The slope of $f(x)$ remains the same and the $y$ - intercept is decreased by 3.
28. The slope of $f(x)$ remains the same and the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ left 0.5 units.
29. $k = 3$, the graph of $g(x)$ is a vertical shift of the graph of $f(x)$ up 3 units.
30. $k=-2$, the graph of $g(x)$ is a horizontal shift of the graph of $f(x)$ right 2 units.

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Explanation:

Step1: Recall function - transformation rules

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