Graphing Functions: Completing the Square, Quadratic Functions, Axis of Symmetry, and Intercepts
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?
https://brainmass.com/math/graphs-and-functions/completing-square-quadratic-functions-axis-symmetry-543240
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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