Algebra II: Quadratic Functions

On this page we hope to clear up problems you might have with quadratic functions. We discuss and give examples of how changes in the equation y = f(x) affect the graph of the function.

Symmetry
Even and odd functions
Transformations
Stretching and shrinking
Parabolas
x-intercepts (zeros)
Quiz on quadratic functions

Symmetry

1. A graph is said to be symmetric to the y-axis when (x, y) and (-x, y) are points on the graph. Example: Example figure.

2. A graph is said to be symmetric to the x-axis when (x, y) and (x, -y) are points on the graph. Graphs symmetric to the x-axis are never functions! Example: Example figure.

3. A graph is said to be symmetric to the origin when (x, y) and (-x, -y) are points on the graph. Example: Example figure.

Even and Odd Functions

If a graph is symmetric to the y-axis, it is an even function. If f(x) = f(-x), the function is even.

Example:

1.  Problem: Is f(x) = x^2 even?
   Solution: Substitute -x for x.

             f(-x) = (-x)^2
             f(-x) = x^2
             f(-x) = f(x)
             
             The function is even.

A function is odd when it is symmetric to the origin. If f(-x) = -f(x), the function is odd.

Example:

1.  Problem: Is f(x) = x^3 odd?
   Solution: Substitute -x for x.

             f(-x) = (-x)^3
             f(-x) = -x^3
             f(-x) = -f(x)

             The function is odd.

If a function isn't odd or even, it is considered neither.

Horizontal and Vertical Translations

When you alter a graph, you transform it. If you transform a graph without changing its shape, you translate it. Vertical and horizontal transformations are translations. When y = f(x) + d, shift (translate) the graph of y = f(x) vertically (upward if d > 0, downward if d < 0).

Example:

1.  Problem: Translate y = x^2 upward by 1.
   Solution: You have been asked to shift the graph upward 1.
             Rewrite the equation to do this, and then graph.

             y = x^2 + 1

             The accompanying figure is a graph of
             the solution.

When y = f(x + c), translate the graph of y = f(x) horizontally (left if c > 0, right if c < 0).

Example:

1.  Problem: Sketch the graph of y = |x + 2|.
   Solution: First, graph y = |x|, then shift it to the left two
             places.

             The accompanying figure is a graph
             of the solution.

Stretching and Shrinking

When y = a * f(x), multiply the y values of f(x) by a. Leave the x values alone. This is a vertical stretch (if |a| > 1) or shrink (if |a| < 1).

Example:

1.  Problem: Given is a graph of y = f(x).  Sketch y = 2f(x).
             Accompanying figure.
   Solution: Multiply values of y by 2 and keep the same x
             values.  Then redraw the graph.

             By following this link, you will get
             a graph of the solution.

When y = f(b * x), divide the x values of f(x) by b. Leave the y values alone. This is a horizontal stretch (if |b| < 1) or shrink (if |b| > 1).

Example:

1.  Problem: Given is a graph of y = f(x).  Sketch y = f(.5x).
             Accompanying figure.
   Solution: Divide each x value by .5 (same as multiplying
             by 2).  Keep the same y values, and redraw the graph.

             By following this link, you will get
             a graph of the solution.

Parabolas

Graphs of quadratic functions are called parabolas. The basic graph that you need to know is f(x) = x^2. This example figure depicts that graph.

If you know that graph, all possible transformations and stretches or shrinks will be a breeze! Notice that the graph of f(x) = x^2 is symmetric to the y-axis, and it's an even function.

X-Intercept and Graphs

Points where a graph crosses or touches the x-axis are called x-intercepts or zeros. To find the zeros of a quadratic equation, you set the equation equal to zero and solve for x.

Example:

1.  Problem: Find the zeros of f(x) = x^2 + x - 2.
   Solution: Set the equation equal to zero.

             0 = x^2 + x - 2

             Factor or use the quadratic formula and solve for x.

             0 = (x + 2)(x - 1)

             x = -2, 1

             The zeros occur when x equals -2 and 1.

Take the quiz on quadratic functions. The quiz is very useful for either review or to see if you've really got the topic down.

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