Question

Graph the function \(f\) by starting with the graph of \(y=x^{2}\) and using transformations. \(f(x)=\frac{1}{2} x^{2}+x-1\)

Short answer

Apply a vertical compression by \( \frac{1}{2} \), shift horizontally by adding \( x \), and shift vertically by subtracting 1.

Step-by-step solution

Identify the base function

The base function given is the quadratic function: \[y = x^2\]This is a standard parabola opening upwards with its vertex at the origin \((0, 0)\).

Apply the vertical stretch/compression

The function can be written as \[y = \frac{1}{2}x^2\]This transformation compresses the parabola vertically by a factor of \( \frac{1}{2} \), making it wider.

Apply the horizontal shift

To the function from Step 2, \[y = \frac{1}{2}x^2\], add a linear term: \[y = \frac{1}{2}x^2 + x\].This term shifts the vertex of the parabola horizontally.

Apply the vertical shift

Finally, to the function from Step 3, \[y = \frac{1}{2}x^2 + x\], subtract 1: \[f(x) = \frac{1}{2}x^2 + x - 1\].This shifts the entire graph down by 1 unit.

Summary of transformations

Starting with \[y = x^2\]: 1. Vertically compress by \( \frac{1}{2} \): \[y = \frac{1}{2}x^2\]2. Add \( x \) to shift the vertex horizontally: \[y = \frac{1}{2}x^2 + x\]3. Subtract 1 to shift vertically: \[f(x) = \frac{1}{2}x^2 + x - 1\]

Key concepts

Function Transformations

To understand graphing quadratic functions better, it's important to learn about function transformations. Transformations help us understand how to shift, stretch, compress, or reflect a function to get a new function.

There are four main types of transformations:

• Vertical shifts
• Horizontal shifts
• Vertical compressions or stretches
• Horizontal compressions or stretches

In our exercise, we specifically deal with vertical compression, horizontal shift, and vertical shift. By mastering these transformations, you can graph any quadratic function easily.

Vertical Compression

Vertical compression is when you make a function 'wider' or 'flatter' by multiplying it by a constant between 0 and 1. In mathematical terms, if you have a function y = x^2 and you multiply it by 1/2, it becomes y = 1/2 x^2.

In our exercise, the base function is y = x^2. When we multiply by 1/2, our function is vertically compressed by a factor of 1/2. So, the parabola becomes wider and less steep. Think about a rubber band being stretched horizontally; it becomes wider and thinner, and the same principle applies here.

Horizontal Shift

A horizontal shift moves the graph of a function left or right. In our exercise, after vertically compressing the function, we add an x term to it. This shifts the vertex horizontally.

Initially, our function after vertical compression is y = 1/2 x^2. When we add an x term, it becomes y = 1/2 x^2 + x. This term contributes to moving the graph sideways, though for more precise movement control, one typically uses y = f(x - h) for right and y = f(x + h) for left. Understanding these shifts helps one control where the vertex of the parabola lies along the x-axis.

Vertical Shift

Vertical shifts move the graph of a function up or down along the y-axis. To perform a vertical shift, you either add or subtract a constant from the function.

In our example, adding or subtracting a number directly changes the entire function's vertical position. Our function from Step 3 is y = 1/2 x^2 + x. When we subtract 1, it becomes y = 1/2 x^2 + x - 1. This shifts the entire graph down by one unit, thus finalizing our function transformation.

Combining vertical compression, horizontal shift, and vertical shift, you can understand how to graph any given quadratic function from a basic one like y = x^2.