# Solutions to Problem Set 30

Math 101-06

11-27-2017

1. Graph the quadratic function . Find and label the roots, and find the x and y-coordinates of the vertex.

The roots are and .

The vertex is halfway between the roots, so the x-coordinate of the vertex is . To find the y-coordinate, plug into :

Since the coefficient of is positive, the graph opens upward.

2. Graph the quadratic function . Find and label the roots, and find the x and y-coordinates of the vertex.

The roots are and .

The vertex is halfway between the roots, so the x-coordinate of the vertex is . To find the y-coordinate, plug into :

Since the coefficient of is negative, the graph opens downward.

3. Graph the quadratic function . Find and label the roots, and find the x and y-coordinates of the vertex.

The root is .

The vertex is halfway between the roots, so the x-coordinate of the vertex is . To find the y-coordinate, plug into :

Since the coefficient of is positive, the graph opens upward.

4. Graph the quadratic function . Find and label the roots, and find the x and y-coordinates of the vertex.

The roots are and .

The vertex is halfway between the roots, so the x-coordinate of the vertex is . To find the y-coordinate, plug into :

Since the coefficient of is negative, the graph opens downward.

5. Graph the quadratic function . Find and label the roots, and find the x and y-coordinates of the vertex.

Since doesn't factor, I'll find the roots using the Quadratic Formula:

Since the roots are complex numbers, the graph doesn't cross the x-axis.

I'll use the formula to find the x-coordinate of the vertex; it is

To find the y-coordinate, plug into :

Since the coefficient of is positive, the graph opens upward.

The only way the graph can open upward and not cross the x-axis is if it is "floating" above the x-axis:

We spend our time envying people whom we wouldn't like to be. - Jean Rostand

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