Identify the following to graph quadratic functions:

  • Axis (or Line) of symmetry
  • Maximum or minimum
  • y-intercept
  • any x-intercept(s)

Axis of Symmetry

Before you begin, click on More Images to view Line of Symmetry, Minimum Point, and y-intercept of y= x^2 + 6x + 5.

Remember, the standard form of a quadratic function is

y = ax2 + bx + c, where a ≠ 0.

The formula for the axis of symmetry is

x = .

Find the axis of symmetry for
y= x2 + 6x + 5.
1. Label a and b.
a= 1, because 1 is the coefficient of x2
b= 6, because 6 is the coefficient of x

2. Plug the numbers into the axis of symmetry formula.
x= -(6)

3. Simplify.
x= -6

x= -3

4. Draw the line x= -3. Notice that along this vertical line, every value of x is -3.

Vertex: Maximum or Minimum?

2 quick tips for the vertices:

  • If the coefficient of a is positive, then the graph opens upward.
  • If the coefficient of a is negative, then the graph opens downward.

Because positive 1 is the coefficient of a, the graph opens upward and its vertex is a minimum. The minimum point is a coordinate in the form (x,y). The value of the line of symmetry is always the x-value of the vertex. To find the y-value for the vertex, follow these steps:

1. Plug the x-value into the function.
y= x2 + 6x + 5
y= (-3)2 + 6(-3) + 5

2. Simplify.
y= 9 + -18 + 5
y= -4

3. Plot the minimum, (-3, -4).

Finding the y-intercept

The y-intercept is where the parabola intersects the y-axis. This is also the point at which x=0.

1. Let x=0 in the function and find y.
y= x2 + 6x + 5
y= (0)2 + 6(0) + 5

2. Simplify.
y= 0 + 0 + 5
y= 5

3. Plot the y-intercept, (0, 5).

Use the Quadratic Formula to Find the x-intercepts

Each quadratic function will have two, one, or no x-intercepts.
The most consistent method of finding any x-intercept is the quadratic formula.

Click on More Images for the picture of Quadratic Formula and Graph of x-intercepts. Follow the steps to find any x-intercepts for y= x2 + 6x + 5.

Notice that the x-intercepts are located at (-1,0) and (-5,0).

Final Step (Yes!)

Connect the dots for a perfect parabola. Click on More Images> for the graph of y= x2 + 6x + 5.

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