After reading: What Are Quadratic Functions? , you learned that quadratic functions tell stories. Knowing that a pop star's record sales increased and then decreased is not enough information to graph the data. Get nosy. What was the peak of the record sales? When did record saleshit rock bottom? In other words, quantify the elements of quadratic functions to sketch an accurate graph.
Identify the following to graph quadratic functions:
- Axis (or Line) of symmetry
- Maximum or minimum
- 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.
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
y= 9 + -18 + 5
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
y= 0 + 0 + 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.