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# Graphing Functions: Completing the Square, Quadratic Functions, Axis of Symmetry, and Intercepts

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1. Using completing the square to describe the graph of the following function. Support your answer graphically.
f(x) = -2x^2 + 4x + 1
2. Graph the function: g(x) = (x-2)^3
3. Determine the quadratic function f whose vertex is (3, -2) and passes through (2, 1)
4. Graph the line containing the point P and having slope m: P = (2, -7); m = 0
5. Graph the function and state the vertex, the axis of symmetry, the intercepts, if any: f(x) = x^2-6x+5
What is the vertex?
What is the axis of symmetry?
What are the intercepts, if any?

## SOLUTION This solution is FREE courtesy of BrainMass!

1.

This parabola has a vertex at

We see that as the function approaches , so the vertex point is a maxima (the parabola opens down).
The parabola intersects the x-axis when

Which gives:

It intersects the y-axis when x=0 and this gives:

To summarize:
The parabola has a vertex which is a maxima at so its symmetry axis is x=1
It intersects the x-axis at and
It intersects the y-axis at

It looks like:

2.
The function is

This is a third degree polynomial that has one root with multiplicity 3.
When we have
The graph is the same as that of only transposed 2 units to the right.
It intersects the y-axis when x=0 so the intersection point is
It looks like:

3.
The general form of a quadratic function is:

Where is the vertex.
In our case the vertex is , hence our parabola has the form:

We require that , and this gives for a:

Therefore:

Opening the parenthesis

It looks like:

4.
A line with slope m and y-intercept b has the canonical equation:

In our case m=0, so the line's equation is:

This is an equation of a line parallel to the x axis (y is constant)
Since the line passes through the point (2,-7) we see this constant must be -7, hence the line equation is

And it looks like:

5.
The function is:

We convert it to the canonical form :

We see that the vertex is

And it is a minima (the parabola opens upward)
The function is symmetric about the vertical axis that passes through the vertex:

It intersects the x axis when which gives:

The x-intercepts are and
It intersects the y-axis when x=0 which gives:

The y-intercept is

It looks like:

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