Problem 67
Question
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 transformed function is \(g(x) = \frac{1}{2}(x-5)^2 + 1\).
1Step 1: Identify the Transformation Type
To solve the problem, identify the types of transformations applied to the function \(f(x) = x^2\). It undergoes a vertical compression, a horizontal shift to the right, and a vertical shift upwards.
2Step 2: Apply Vertical Compression
The function is compressed vertically by a factor of \(\frac{1}{2}\). This means that each output value of the original function \(f(x)\) is multiplied by \(\frac{1}{2}\). Therefore, the new function becomes \(f(x) = \frac{1}{2}x^2\).
3Step 3: Apply Horizontal Shift
Next, the graph is shifted to the right by 5 units. To achieve this, replace \(x\) with \((x-5)\) in the function. The function thus transforms into \(\frac{1}{2}(x-5)^2\).
4Step 4: Apply Vertical Shift
Finally, shift the graph upwards by 1 unit by adding 1 to the function. The new function becomes \(g(x) = \frac{1}{2}(x-5)^2 + 1\).
5Step 5: Combine All Transformations
Combine all the transformations into the final function. Therefore, the transformed function is \(g(x) = \frac{1}{2}(x-5)^2 + 1\), where the original function has been vertically compressed, horizontally shifted, and vertically shifted.
Key Concepts
Vertical CompressionHorizontal ShiftVertical Shift
Vertical Compression
In the world of graph transformations, vertical compression is a powerful tool. It involves compressing the graph of a function along the vertical axis. When a function undergoes a vertical compression, each point on the graph is pushed closer to the x-axis. This occurs when you multiply the entire function by a constant that is between 0 and 1.
For example, consider the function \(f(x) = x^2\). If we apply a vertical compression by a factor of \(\frac{1}{2}\), we take every y-value of the original function and multiply it by \(\frac{1}{2}\).
This results in \(g(x) = \frac{1}{2}x^2\).
For example, consider the function \(f(x) = x^2\). If we apply a vertical compression by a factor of \(\frac{1}{2}\), we take every y-value of the original function and multiply it by \(\frac{1}{2}\).
This results in \(g(x) = \frac{1}{2}x^2\).
- The graph retains its shape, but it becomes "flatter" due to the reduction in height.
- As the compressed factor gets smaller, the graph appears more compressed towards the x-axis.
Horizontal Shift
A horizontal shift involves moving the graph left or right along the x-axis. This type of transformation is achieved by adding or subtracting a constant value inside the function's argument. Specifically, if you subtract a positive number from x, the graph shifts to the right.
Conversely, adding a number shifts it to the left.
In our example, the function \(f(x) = \frac{1}{2}x^2\) is shifted to the right by 5 units. This is accomplished by replacing \(x\) with \((x-5)\), resulting in the transformed function \(\frac{1}{2}(x-5)^2\).
Conversely, adding a number shifts it to the left.
In our example, the function \(f(x) = \frac{1}{2}x^2\) is shifted to the right by 5 units. This is accomplished by replacing \(x\) with \((x-5)\), resulting in the transformed function \(\frac{1}{2}(x-5)^2\).
- Moving the graph does not change its shape or orientation.
- This shift only influences the position of the graph, impacting the function's appearance on the x-axis.
Vertical Shift
Vertical shifts, both upwards and downwards, allow for repositioning the graph along the y-axis. This transformation does not alter the shape of the graph or its steepness. It's simply about moving the graph upwards or downwards by adding or subtracting a constant from the function.
In the transformation process for \(g(x)\), after the vertical compression and horizontal shift applied to \(f(x)\), we shift the entire graph upwards by 1 unit. By adding 1 to our transformed function \(\frac{1}{2}(x-5)^2\), we achieve \(g(x) = \frac{1}{2}(x-5)^2 + 1\).
In the transformation process for \(g(x)\), after the vertical compression and horizontal shift applied to \(f(x)\), we shift the entire graph upwards by 1 unit. By adding 1 to our transformed function \(\frac{1}{2}(x-5)^2\), we achieve \(g(x) = \frac{1}{2}(x-5)^2 + 1\).
- The graph moves up by 1 unit, causing every point on the graph to increase by 1 in the y-value.
- Vertical shifts maintain the orientation and curvature of the original function.
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Problem 67
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