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By the end of this section, you will be able to:
  • Recognize the graph of a quadratic equation in two variables
  • Find the axis of symmetry and vertex of a parabola
  • Find the intercepts of a parabola
  • Graph quadratic equations in two variables
  • Solve maximum and minimum applications

Before you get started, take this readiness quiz.

  1. Graph the equation y = 3 x 5 by plotting points.
    If you missed this problem, review [link] .
  2. Evaluate 2 x 2 + 4 x 1 when x = −3 .
    If you missed this problem, review [link] .
  3. Evaluate b 2 a when a = 1 3 and b = 5 6 .
    If you missed this problem, review [link] .

Recognize the graph of a quadratic equation in two variables

We have graphed equations of the form A x + B y = C . We called equations like this linear equations because their graphs are straight lines.

Now, we will graph equations of the form y = a x 2 + b x + c . We call this kind of equation a quadratic equation in two variables    .

Quadratic equation in two variables

A quadratic equation in two variables    , where a , b , and c are real numbers and a 0 , is an equation of the form

y = a x 2 + b x + c

Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.

Let’s look first at graphing the quadratic equation y = x 2 . We will choose integer values of x between −2 and 2 and find their y values. See [link] .

y = x 2
x y
0 0
1 1
−1 1
2 4
−2 4

Notice when we let x = 1 and x = −1 , we got the same value for y .

y = x 2 y = x 2 y = 1 2 y = ( −1 ) 2 y = 1 y = 1

The same thing happened when we let x = 2 and x = −2 .

Now, we will plot the points to show the graph of y = x 2 . See [link] .

This figure shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (0, 0). Other points on the curve are located at (-2, 4), (-1, 1), (1, 1) and (2, 4).

The graph is not a line. This figure is called a parabola    . Every quadratic equation has a graph that looks like this.

In [link] you will practice graphing a parabola by plotting a few points.

Graph y = x 2 1 .

Solution

We will graph the equation by plotting points.

Choose integers values for x , substitute them into the equation and solve for y .
Record the values of the ordered pairs in the chart. .
Plot the points, and then connect them with a smooth curve. The result will be the graph of the equation y = x 2 1 . .
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How do the equations y = x 2 and y = x 2 1 differ? What is the difference between their graphs? How are their graphs the same?

All parabolas of the form y = a x 2 + b x + c open upwards or downwards. See [link] .

This figure shows two graphs side by side. The graph on the left side shows an upward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The lowest point on the curve is at the point (-2, -1). Other points on the curve are located at (-3, 0), and (-1, 0). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the inequality a greater than 0 which means the parabola opens upwards. The graph on the right side shows a downward-opening u shaped curve graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The highest point on the curve is at the point (2, 7). Other points on the curve are located at (0, 3), and (4, 3). Below the graph is the equation y equals a squared plus b x plus c. Below that is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the inequality a less than 0 which means the parabola opens downwards.

Notice that the only difference in the two equations is the negative sign before the x 2 in the equation of the second graph in [link] . When the x 2 term is positive, the parabola    opens upward, and when the x 2 term is negative, the parabola opens downward.

Parabola orientation

For the quadratic equation y = a x 2 + b x + c , if:

The image shows two statements. The first statement reads “a greater than 0, the parabola opens upwards”. This statement is followed by the image of an upward opening parabola. The second statement reads “a less than 0, the parabola opens downward”. This statement is followed by the image of a downward opening parabola.

Determine whether each parabola opens upward or downward:

y = −3 x 2 + 2 x 4 y = 6 x 2 + 7 x 9

Solution


Find the value of " a ".
.
Since the “a” is negative, the parabola will open downward.

Find the value of " a ".
.
Since the “a” is positive, the parabola will open upward.

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Determine whether each parabola opens upward or downward:

y = 2 x 2 + 5 x 2 y = −3 x 2 4 x + 7

up down

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Determine whether each parabola opens upward or downward:

y = −2 x 2 2 x 3 y = 5 x 2 2 x 1

down up

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Find the axis of symmetry and vertex of a parabola

Look again at [link] . Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry    of the parabola.

Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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