Problem 13 - Algebra 2

Question

Graph each function as a transformation of its parent function. $$ y=52\left(\frac{2}{13}\right)^{x-1}+26 $$

Step-by-Step Solution

Answer

The graph of \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \) would start at the y-value of 26 on the y-axis (due to the +26 vertical shift), be trending downwards (as the base of the exponential is less than 1) and be stretched by a factor of 52 vertically and shifted 1 unit to the right. The graph represents a vertically stretched and horizontally flipped exponential decay translated to the right and upwards.

Step 1: Identify the transformations

First, recognize the transformations present in the function. These are given by the terms added or multiplied to the function. Specifically, 52 is the vertical stretch, \( \frac{2}{13} \) is the base of the exponential indicating a horizontal flip and vertical shrink since it’s less than 1, x-1 translates the graph 1 unit to the right, and +26 translates the graph upward by 26 units.

Step 2: Graph the parent function

Next, graph the parent function, \( y=b^x \), and apply transformations one by one. Keep in mind that the parent function \( y=b^x \) passes through point (0,1) and is increasing for \( b>1 \) and decreasing for \( 0

Step 3: Apply the transformations

Since the base of the exponential is \( \frac{2}{13} \), it indicates a horizontal flip and vertical shrink, so the graph shape changes accordingly. The factor of 52 vertically stretches the graph, so all y-values should be 52 times their initial values. After, apply the horizontal and vertical shifts. The term (x - 1) translates the graph 1 unit to the right, and +26 translates the graph upward by 26 units.

Step 4: Review the graph

After applying the transformations, review the graph to ensure that it accurately represents the function. It should cross the y-axis at 26 and be decreasing due to the base \( \frac{2}{13} \) being less than 1.

Key Concepts

Parent Function

The concept of a "Parent Function" is central to understanding how transformations alter a graph. A parent function acts as a starting point or a basic blueprint from which more complex functions are derived. For exponential functions, the simplest form of the parent function is \( y = b^x \), where \( b \) is the base. This function pass through the point (0,1), indicating its vertical starting point is 1 when \( x = 0 \). The behavior and shape of this parent function depend on the value of \( b \):

• If \( b > 1 \), the function is increasing.
• If \( 0 < b < 1 \), the function is decreasing.

Starting from the parent function allows us to clearly see how the various transformations adjust the graph's appearance.

Transformations

Transformations are modifications made to the parent function to obtain a new graph. They allow us to alter the graph's position, shape, or size. In the function \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \), several transformations occur:

• A horizontal shift, due to \( x-1 \).
• A vertical stretch by a factor of 52.
• A vertical shift upwards by adding 26.
• A reflection and shrink, due to the base \( \frac{2}{13} \).

Each transformation can influence the graph independently, allowing us to visually map how each change affects the function's position, size, and orientation.

Vertical Stretch

Vertical stretch involves multiplying the output of a function by a constant, changing the function's range and making it "taller." In the function \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \), the term 52 is responsible for the vertical stretch. It means we take the y-values from the parent function and multiply them by 52, stretching the graph vertically.
After a vertical stretch, every point moves farther away from the x-axis except those on the axis itself. This transformation amplifies the function’s output, making features such as peaks and troughs more discernible.

Horizontal Shift

Horizontal shifts move the graph to the left or right along the x-axis. They occur due to transformations on the input variable \( x \). In our equation \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \), the expression \( x-1 \) signifies a horizontal shift to the right by 1 unit.
This shift is simple to apply: revise each x-coordinate in the graph by adding 1 to its value. This transformation solely affects the x-values, so it adjusts the graph's position without changing its shape. When analyzing graphs, understanding horizontal shifts is vital, allowing us to anticipate the direction in which the function is translated.

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Other exercises in this chapter

Problem 12

Write an exponential function \(y=a b^{x}\) for a graph that includes the given points. $$ (2,18),(5,60.75) $$

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Problem 13

Solve by graphing. Round to the nearest ten-thousandth. $$ 4^{7 x}=250 $$

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Problem 13

Write each logarithmic expression as a single logarithm. \(5 \log 3+\log 4\)

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Problem 13

Write each equation in logarithmic form. $$ 10^{-2}=0.01 $$

View solution

Key Concepts

Parent Function

The concept of a "Parent Function" is central to understanding how transformations alter a graph. A parent function acts as a starting point or a basic blueprint from which more complex functions are derived. For exponential functions, the simplest form of the parent function is \( y = b^x \), where \( b \) is the base. This function pass through the point (0,1), indicating its vertical starting point is 1 when \( x = 0 \). The behavior and shape of this parent function depend on the value of \( b \):

• If \( b > 1 \), the function is increasing.
• If \( 0 < b < 1 \), the function is decreasing.

Starting from the parent function allows us to clearly see how the various transformations adjust the graph's appearance.

Transformations

Transformations are modifications made to the parent function to obtain a new graph. They allow us to alter the graph's position, shape, or size. In the function \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \), several transformations occur:

• A horizontal shift, due to \( x-1 \).
• A vertical stretch by a factor of 52.
• A vertical shift upwards by adding 26.
• A reflection and shrink, due to the base \( \frac{2}{13} \).

Each transformation can influence the graph independently, allowing us to visually map how each change affects the function's position, size, and orientation.

Vertical Stretch

Vertical stretch involves multiplying the output of a function by a constant, changing the function's range and making it "taller." In the function \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \), the term 52 is responsible for the vertical stretch. It means we take the y-values from the parent function and multiply them by 52, stretching the graph vertically.
After a vertical stretch, every point moves farther away from the x-axis except those on the axis itself. This transformation amplifies the function’s output, making features such as peaks and troughs more discernible.

Horizontal Shift

Horizontal shifts move the graph to the left or right along the x-axis. They occur due to transformations on the input variable \( x \). In our equation \( y=52\left(\frac{2}{13}\right)^{x-1}+26 \), the expression \( x-1 \) signifies a horizontal shift to the right by 1 unit.
This shift is simple to apply: revise each x-coordinate in the graph by adding 1 to its value. This transformation solely affects the x-values, so it adjusts the graph's position without changing its shape. When analyzing graphs, understanding horizontal shifts is vital, allowing us to anticipate the direction in which the function is translated.