Question

A quadratic equation can have either 2, 1, or 0 solutions. Describe what the graph would look like for each case.

A function f(x) is shifted by the following transformation, f(x - 2) + 1. Explain how the original graph changes.

Answer 1

This is really two different questions so I'll answer them one at a time.

First Question: A quadratic equation can have either 2, 1, or 0 solutions. Describe what the graph would look like for each case.

An easy way to think about this is "The number of solutions of a quadratic equation is the number of times the graph of the equation touches the x-axis." (NOTE: This is why you will sometimes see solutions called "zeros": When the graph is touching the x-axis, y=0.)

2 solutions:


1 solution:


0 solutions:



Second Question:A function f(x) is shifted by the following transformation, f(x - 2) + 1. Explain how the original graph changes

This function has two shifts being applied to it.
The first shift is of the type:
\[f(x) \rightarrow f(x+a)\]
This type of shift moves the graph left or right. The graph moves left if 'a' is positive and right if 'a' is negative.

The second shift is of the type:
\[f(x) \rightarrow f(x)+b\]
This type of shift moves the graph up or down. The graph moves up if 'b' is positive and down if 'b' is negative.

Using these two rules we can see that the graph of the function has been moved right 2 units (a=-2) and up one unity (b=+1)