Math / Algebra

QuestionGiven the function $y=x^{3}$ , what are the parameters of the transformed function $y=\frac{1}{5}(x+2)^{3}+9$ and what is the effect of each parameter on the graph of the original function? $a=\frac{1}{5}$ , vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ $h=-2$ , horizontal translation 2 units left $k=9$ , vertical translation 9 units up $a=5$ , vertical stretch about the $x$ -axis by a factor of 5 $h=-2$ , horizontal translation 2 units left $k=9$ , vertical translation 9 units right $a=5$ , vertical stretch about the $x$ -axis by a factor of 5 $h=-2$ , horizontal translation 2 units right $k=9$ , vertical translation 9 units down $a=\frac{1}{5}$ , vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ $h=9$ , horizontal translation 9 units left $k=-2$ , vertical translation 2 units down

Studdy Solution

STEP 1

1. The original function is $y = x^3$ .
2. The transformed function is $y = \frac{1}{5}(x+2)^3 + 9$ .
3. We need to identify parameters $a$ , $h$ , and $k$ in the transformed function and their effects on the graph.

STEP 2

1. Identify the parameter $a$ and its effect.
2. Identify the parameter $h$ and its effect.
3. Identify the parameter $k$ and its effect.
4. Match the identified parameters and effects with the given options.

STEP 3

Identify the parameter $a$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $a$ is the coefficient of the cubic term, which is $\frac{1}{5}$ .
Effect: This represents a vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ .

STEP 4

Identify the parameter $h$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $h$ is the horizontal shift, found inside the parentheses with $x$ . Here, $h = -2$ .
Effect: This represents a horizontal translation 2 units left.

STEP 5

Identify the parameter $k$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $k$ is the constant added outside the cubic term, which is $9$ .
Effect: This represents a vertical translation 9 units up.

STEP 6

Match the identified parameters and effects with the given options:
- $a = \frac{1}{5}$ , vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ - $h = -2$ , horizontal translation 2 units left - $k = 9$ , vertical translation 9 units up
This matches with the first option provided.
The parameters and their effects are: - $a = \frac{1}{5}$ , vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ - $h = -2$ , horizontal translation 2 units left - $k = 9$ , vertical translation 9 units up

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Studdy Solution

STEP 1

1. The original function is $y = x^3$ .
2. The transformed function is $y = \frac{1}{5}(x+2)^3 + 9$ .
3. We need to identify parameters $a$ , $h$ , and $k$ in the transformed function and their effects on the graph.

STEP 2

1. Identify the parameter $a$ and its effect.
2. Identify the parameter $h$ and its effect.
3. Identify the parameter $k$ and its effect.
4. Match the identified parameters and effects with the given options.

STEP 3

Identify the parameter $a$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $a$ is the coefficient of the cubic term, which is $\frac{1}{5}$ .
Effect: This represents a vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ .

STEP 4

Identify the parameter $h$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $h$ is the horizontal shift, found inside the parentheses with $x$ . Here, $h = -2$ .
Effect: This represents a horizontal translation 2 units left.

STEP 5

Identify the parameter $k$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $k$ is the constant added outside the cubic term, which is $9$ .
Effect: This represents a vertical translation 9 units up.

STEP 6

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Studdy Solution

STEP 1

1. The original function is y=x3 y = x^3 y=x3.2. The transformed function is y=15(x+2)3+9 y = \frac{1}{5}(x+2)^3 + 9 y=51​(x+2)3+9.3. We need to identify parameters a a a, h h h, and k k k in the transformed function and their effects on the graph.

STEP 2

1. Identify the parameter a a a and its effect.2. Identify the parameter h h h and its effect.3. Identify the parameter k k k and its effect.4. Match the identified parameters and effects with the given options.

STEP 3

Identify the parameter a a a in the transformed function y=15(x+2)3+9 y = \frac{1}{5}(x+2)^3 + 9 y=51​(x+2)3+9.The parameter a a a is the coefficient of the cubic term, which is 15\frac{1}{5}51​.Effect: This represents a vertical stretch about the x x x-axis by a factor of 15\frac{1}{5}51​.

STEP 4

Identify the parameter h h h in the transformed function y=15(x+2)3+9 y = \frac{1}{5}(x+2)^3 + 9 y=51​(x+2)3+9.The parameter h h h is the horizontal shift, found inside the parentheses with x x x. Here, h=−2 h = -2 h=−2.Effect: This represents a horizontal translation 2 units left.

STEP 5

Identify the parameter k k k in the transformed function y=15(x+2)3+9 y = \frac{1}{5}(x+2)^3 + 9 y=51​(x+2)3+9.The parameter k k k is the constant added outside the cubic term, which is 9 9 9.Effect: This represents a vertical translation 9 units up.

STEP 6

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1. The original function is $y = x^3$ .
2. The transformed function is $y = \frac{1}{5}(x+2)^3 + 9$ .
3. We need to identify parameters $a$ , $h$ , and $k$ in the transformed function and their effects on the graph.

1. Identify the parameter $a$ and its effect.
2. Identify the parameter $h$ and its effect.
3. Identify the parameter $k$ and its effect.
4. Match the identified parameters and effects with the given options.

Identify the parameter $a$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $a$ is the coefficient of the cubic term, which is $\frac{1}{5}$ .
Effect: This represents a vertical stretch about the $x$ -axis by a factor of $\frac{1}{5}$ .

Identify the parameter $h$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $h$ is the horizontal shift, found inside the parentheses with $x$ . Here, $h = -2$ .
Effect: This represents a horizontal translation 2 units left.

Identify the parameter $k$ in the transformed function $y = \frac{1}{5}(x+2)^3 + 9$ .
The parameter $k$ is the constant added outside the cubic term, which is $9$ .
Effect: This represents a vertical translation 9 units up.