Problem 14 - Algebra 2

Question

Graph each function as a transformation of its parent function. $$ y=9\left(\frac{1}{3}\right)^{x+7}-3 $$

Step-by-Step Solution

Answer

The graph of the function \(y = 9(1/3)^{x+7} - 3\) is a graph of the parent function \(y = (1/3)^x\), vertically stretched by a factor of 9, shifted 7 units to the left and 3 units down.

Step 1: Identify the components of the function

In the function \(y = 9(1/3)^{x + 7} - 3\), the value of \(a\) is 9, which is a vertical stretch by a factor of 9, the base \(b\) of the exponential part is 1/3, the \(h\)-value is -7, indicating a shift 7 units to the left, and the \(k\)-value is -3, indicating a downward shift by 3 units.

Step 2: Graph the parent function

The parent function here is \(y = b^x\) where \(b = 1/3\). Begin by first plotting the function \(y = (1/3)^x\). Considering that \(y = 1\) when \(x = 0\), it is evident that the function decreases as \(x\) increases.

Step 3: Apply the vertical stretch

Next, apply the vertical stretch by multiplying the \(y\)-values of each point of the parent function by \(a = 9\). This raises the curve of the function.

Step 4: Apply the horizontal shift

Shift the function to the left by \(h = 7\) units. To do so, subtract 7 from each \(x\)-value.

Step 5: Apply the vertical shift

Lastly, shift the function downward by \(k = 3\) units. To do so, subtract 3 from each \(y\)-value.

Key Concepts

Exponential Functions

Exponential functions are a core component of many mathematical concepts. An exponential function is a type of function where a constant base is raised to a variable exponent. This means the function has the form \( y = a \, b^{x} + c \), where:\

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• \( a \) is a coefficient that can stretch or compress the function vertically.
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• \( b \) is the base of the exponent and typically greater than zero.
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• \( x \) is the exponent, which is a variable.
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• \( c \) is a constant that can shift the entire function vertically.
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\In the equation provided, \( b = \frac{1}{3} \), which impacts how the function behaves as \( x \) changes. Since \( \frac{1}{3} \) is between 0 and 1, the graph of \( y = \left(\frac{1}{3}\right)^x \) represents a decreasing exponential function. This type of graph typically starts high on the y-axis and descends toward 0 but never quite reaches it. This behavior is crucial for understanding transformations in exponential functions.

Vertical Stretch and Shift

Vertical transformations affect how steep or shallow the graph of a function is and where it is located vertically. A vertical stretch or compression is determined by the coefficient \( a \) in the function. If \( a \) is a number greater than 1, the graph stretches vertically. If \( a \) is between 0 and 1, the graph compresses. In our function, \( a = 9 \), meaning the graph is stretched vertically by a factor of 9.\
\Additionally, the constant \( k \) in the function affects vertical shifting. A positive \( k \) shifts the graph up, and a negative \( k \) shifts it down. Here, \( k = -3 \) shifts the entire function downward by 3 units. To visualize this, imagine the entire graph moving down by 3 units alongside multiplying all y-values by 9, making the graph steeper.

Horizontal Shift

Horizontal shifts change the position of the graph along the x-axis. This is influenced by the variable \( h \) in the expression \( x + h \) within the exponent. The shift's direction is counterintuitive: if \( h \) is positive, the graph moves to the left; if \( h \) is negative, it moves to the right. In our example, \( x + 7 \) means the function moves 7 units left.\
\To apply this concept, adjust the x-coordinates of each point along the graph of the parent function \( y = b^x \). Moving each point 7 units left is equivalent to subtracting 7 from every x-value. This transformation will then lead to the new position of the exponential graph aligning accordingly without altering its shape. Understanding this shift is critical for accurately plotting the transformed function.