## Example Questions

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### Example Question #1 : How To Graph A Quadratic Function What is the equation of the graph?      Explanation:

In order to figure out what the equation of the image is, we need to find the vertex. From the graph we can determine that the vertex is at . We can use vertex form to solve for the equation of this graph.

Recall vertex form, , where is the coordinate of the vertex, and is the coordinate of the vertex.

Plugging in our values, we get To solve for , we need to pick a point on the graph and plug it into our equation.

I will pick .     Now our equation is Let's expand this,   ### Example Question #2 : How To Graph A Quadratic Function

How many times does the equation below cross the x-axis?      Explanation:

You can solve this problem two ways.

3. You can solve for where the graph crosses the x-axis by setting the equation equal to zero, factoring, and solving.     2. You can quickly sketch the graph by choosing some x values and solving for y.  We see that the graph passes the x-axis twice.

### Example Question #3 : How To Graph A Quadratic Function

Let f(x) = x2. By how many units must f(x) be shifted downward so that the distance between its x-intercepts becomes 8?

2

8

12

16

4

16

Explanation:

Because the graph of f(x) = x2 is symmetric about the y-axis, when we shift it downward, the points where it intersects the x-axis will be the same distance from the origin. In other words, we could say that one intercept will be (-a,0) and the other would be (a,0). The distance between these two points has to be 8, so that means that 2a = 8, and a = 4. This means that when f(x) is shifted downward, its new roots will be at (-4,0) and (4,0).

Let g(x) be the graph after f(x) has been shifted downward. We know that g(x) must have the roots (-4,0) and (4,0). We could thus write the equation of g(x) as (x-(-4))(x-4) = (x+4)(x-4) = x2 - 16.

We can now compare f(x) and g(x), and we see that g(x) could be obtained if f(x) were shifted down by 16 units; therefore, the answer is 16.

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