Math / Algebra

QuestionTransformation Puzzler - Transforming Quadratic Functions The quadratic parent function, $a(x)=x^{2}$ , was transformed nine times to end up with the function $j(x)=-1 / 2(x+5)^{2}$ . These transformations are shown in the table below. Follow each step and describe what transformation occurred to the graph of each function to end up with the resulting function. The first example has been done for you as reference. reate your own function starting with $f(x)$ to show $g(x)=2 f(x+3)-4$

Studdy Solution

STEP 1

1. We are transforming the quadratic parent function $a(x) = x^2$ to $j(x) = -\frac{1}{2}(x+5)^2$ .
2. Each transformation corresponds to a specific change in the function's graph.
3. The goal is to understand each transformation step and apply a similar process to create the function $g(x) = 2f(x+3) - 4$ .

STEP 2

1. Identify and describe the transformations applied to $a(x) = x^2$ to obtain $j(x) = -\frac{1}{2}(x+5)^2$ .
2. Apply similar transformation principles to create the function $g(x) = 2f(x+3) - 4$ .

STEP 3

Start with the parent function $a(x) = x^2$ .

STEP 4

Identify the horizontal shift: The function $(x+5)^2$ indicates a horizontal shift to the left by 5 units.

STEP 5

Identify the vertical stretch/compression and reflection: The coefficient $-\frac{1}{2}$ indicates a vertical compression by a factor of $\frac{1}{2}$ and a reflection over the x-axis.

STEP 6

Start with the function $f(x)$ .

STEP 7

Apply the horizontal shift: The term $f(x+3)$ indicates a horizontal shift to the left by 3 units.

STEP 8

Apply the vertical stretch and shift: The term $2f(x+3)$ indicates a vertical stretch by a factor of 2, and subtracting 4 indicates a vertical shift downward by 4 units.
The function $g(x) = 2f(x+3) - 4$ is obtained by applying these transformations.

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Studdy Solution

STEP 1

STEP 2

1. Identify and describe the transformations applied to $a(x) = x^2$ to obtain $j(x) = -\frac{1}{2}(x+5)^2$ .
2. Apply similar transformation principles to create the function $g(x) = 2f(x+3) - 4$ .

STEP 3

Start with the parent function $a(x) = x^2$ .

STEP 4

Identify the horizontal shift: The function $(x+5)^2$ indicates a horizontal shift to the left by 5 units.

STEP 5

Identify the vertical stretch/compression and reflection: The coefficient $-\frac{1}{2}$ indicates a vertical compression by a factor of $\frac{1}{2}$ and a reflection over the x-axis.

STEP 6

Start with the function $f(x)$ .

STEP 7

Apply the horizontal shift: The term $f(x+3)$ indicates a horizontal shift to the left by 3 units.

STEP 8

Apply the vertical stretch and shift: The term $2f(x+3)$ indicates a vertical stretch by a factor of 2, and subtracting 4 indicates a vertical shift downward by 4 units.
The function $g(x) = 2f(x+3) - 4$ is obtained by applying these transformations.

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Studdy Solution

STEP 1

STEP 2

1. Identify and describe the transformations applied to a(x)=x2 a(x) = x^2 a(x)=x2 to obtain j(x)=−12(x+5)2 j(x) = -\frac{1}{2}(x+5)^2 j(x)=−21​(x+5)2.2. Apply similar transformation principles to create the function g(x)=2f(x+3)−4 g(x) = 2f(x+3) - 4 g(x)=2f(x+3)−4.

STEP 3

Start with the parent function a(x)=x2 a(x) = x^2 a(x)=x2.

STEP 4

Identify the horizontal shift: The function (x+5)2 (x+5)^2 (x+5)2 indicates a horizontal shift to the left by 5 units.

STEP 5

Identify the vertical stretch/compression and reflection: The coefficient −12-\frac{1}{2}−21​ indicates a vertical compression by a factor of 12\frac{1}{2}21​ and a reflection over the x-axis.

STEP 6

Start with the function f(x) f(x) f(x).

STEP 7

Apply the horizontal shift: The term f(x+3) f(x+3) f(x+3) indicates a horizontal shift to the left by 3 units.

STEP 8

Apply the vertical stretch and shift: The term 2f(x+3) 2f(x+3) 2f(x+3) indicates a vertical stretch by a factor of 2, and subtracting 4 indicates a vertical shift downward by 4 units. The function g(x)=2f(x+3)−4 g(x) = 2f(x+3) - 4 g(x)=2f(x+3)−4 is obtained by applying these transformations.

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1. Identify and describe the transformations applied to $a(x) = x^2$ to obtain $j(x) = -\frac{1}{2}(x+5)^2$ .
2. Apply similar transformation principles to create the function $g(x) = 2f(x+3) - 4$ .

Start with the parent function $a(x) = x^2$ .

Identify the horizontal shift: The function $(x+5)^2$ indicates a horizontal shift to the left by 5 units.

Identify the vertical stretch/compression and reflection: The coefficient $-\frac{1}{2}$ indicates a vertical compression by a factor of $\frac{1}{2}$ and a reflection over the x-axis.

Start with the function $f(x)$ .

Apply the horizontal shift: The term $f(x+3)$ indicates a horizontal shift to the left by 3 units.

Apply the vertical stretch and shift: The term $2f(x+3)$ indicates a vertical stretch by a factor of 2, and subtracting 4 indicates a vertical shift downward by 4 units.
The function $g(x) = 2f(x+3) - 4$ is obtained by applying these transformations.