Problem 67
Question
For the following exercises, write a formula for the function \(g\) that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=x^{2}\) is vertically compressed by a factor of \(\frac{1}{2},\) then shifted to the right 5 units and up 1 unit.
Step-by-Step Solution
Verified Answer
The function is \(g(x) = \frac{1}{2}(x-5)^2 + 1\).
1Step 1: Understand the Initial Function
The initial function given is \(f(x) = x^2\). This is a standard quadratic function with a vertex at the origin and a parabola opening upwards.
2Step 2: Apply the Vertical Compression
A vertical compression by a factor of \(\frac{1}{2}\) means we multiply the function by \(\frac{1}{2}\). So, we modify the function to \(g(x) = \frac{1}{2}x^2\).
3Step 3: Shift the Graph Right by 5 Units
Shifting the graph to the right by 5 units involves replacing \(x\) with \(x-5\). The function modifies to \(g(x) = \frac{1}{2}(x-5)^2\).
4Step 4: Shift the Graph Up by 1 Unit
Now, shift the graph up by adding 1 to the function. Thus, \(g(x) = \frac{1}{2}(x-5)^2 + 1\).
Key Concepts
Function TransformationVertical CompressionHorizontal ShiftVertical Shift
Function Transformation
Function transformation involves changing the position or shape of a graph in relation to the axes, usually by altering its equation.
For quadratic functions like \(f(x) = x^2\), transformations can include operations like translations, scaling, and reflections.
Changing a function can visually represent it differently on a graph, without changing its fundamental nature.
Many transformations can be applied to a function, including:
For quadratic functions like \(f(x) = x^2\), transformations can include operations like translations, scaling, and reflections.
Changing a function can visually represent it differently on a graph, without changing its fundamental nature.
Many transformations can be applied to a function, including:
- Vertical shifts (moving the graph up or down)
- Horizontal shifts (moving the graph left or right)
- Vertical and horizontal scaling (compressing or stretching the graph)
- Reflections (flipping the graph over the x-axis or y-axis)
Vertical Compression
Vertical compression affects the 'height' of the graph of a function. It makes the graph "shorter" by reducing the y-values of the points on the graph.
When a quadratic function like \(f(x) = x^2\) is vertically compressed, each y-value is reduced by a certain factor, decreasing the steepness of the graph.
In mathematical terms, vertical compression by a factor of \(\frac{1}{2}\) involves multiplying the function output by \(\frac{1}{2}\).
For example, transforming \(f(x) = x^2\) with a compression gives us \(g(x) = \frac{1}{2}x^2\).
This modifies the graph of x² to become less steep and closer to the x-axis.
Vertical compression is useful for analyzing how a function’s rate of change or steepness can be modified.
When a quadratic function like \(f(x) = x^2\) is vertically compressed, each y-value is reduced by a certain factor, decreasing the steepness of the graph.
In mathematical terms, vertical compression by a factor of \(\frac{1}{2}\) involves multiplying the function output by \(\frac{1}{2}\).
For example, transforming \(f(x) = x^2\) with a compression gives us \(g(x) = \frac{1}{2}x^2\).
This modifies the graph of x² to become less steep and closer to the x-axis.
Vertical compression is useful for analyzing how a function’s rate of change or steepness can be modified.
Horizontal Shift
A horizontal shift moves the graph of a function left or right along the x-axis.
To shift a graph horizontally, the variable x is replaced with \((x-h)\) or \((x+h)\), where 'h' is the number of units to shift.
A rightward shift involves \(x-h\). Thus, for the function \(g(x) = \frac{1}{2}(x-5)^2\), '5' indicates a shift to the right by 5 units.
This means every point on the graph moves to the right by 5 units. The transformation does not affect the shape of the parabola, only its position.
Understanding horizontal shifts helps in predicting the new location of the graph on the coordinate plane.
To shift a graph horizontally, the variable x is replaced with \((x-h)\) or \((x+h)\), where 'h' is the number of units to shift.
A rightward shift involves \(x-h\). Thus, for the function \(g(x) = \frac{1}{2}(x-5)^2\), '5' indicates a shift to the right by 5 units.
This means every point on the graph moves to the right by 5 units. The transformation does not affect the shape of the parabola, only its position.
Understanding horizontal shifts helps in predicting the new location of the graph on the coordinate plane.
Vertical Shift
Vertical shift involves moving a graph up or down along the y-axis.
This is done by adding or subtracting a constant from the function.
Specifically, if \(g(x) = \frac{1}{2}(x-5)^2 + 1\), the '+1' represents a shift of 1 unit upward.
The entire graph is elevated by one unit, moving every point on the curve higher without changing the width or direction of the parabola.
These changes are fundamental in various applications, like adjusting data to fit a model or aligning multiple graphs for comparison.
Vertical shifts enhance our flexibility in manipulating the position of graphs.
This is done by adding or subtracting a constant from the function.
Specifically, if \(g(x) = \frac{1}{2}(x-5)^2 + 1\), the '+1' represents a shift of 1 unit upward.
The entire graph is elevated by one unit, moving every point on the curve higher without changing the width or direction of the parabola.
These changes are fundamental in various applications, like adjusting data to fit a model or aligning multiple graphs for comparison.
Vertical shifts enhance our flexibility in manipulating the position of graphs.
Other exercises in this chapter
Problem 66
For the following exercises, use the function \(f\) represented in Table 14 below. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 &
View solution Problem 67
For the following exercises, use the function values for \(f\) and \(g\) shown in Table 4 to evaluate the expressions. $$\begin{array}{||cc|c|c|c|c|c|c|c|} \hli
View solution Problem 67
Write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=x^{2}\) is verticall
View solution Problem 67
For the following exercises, use the function \(f\) represented in Table 14 below. $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & 0 & 1 & 2 &
View solution