Print this Page
3.7 – Graphs of Quadratic Functions
Objectives
 Determine the number of times that a given function crosses the xaxis.
 Find the vertex and xintercepts of the graph of a quadratic function.
 Use the quadratic formula to find the vertex of a given function.
 Determine if a discriminant is positive, negative, or zero when given the graph of the function.
 Match a graph to its quadratic equation.
Key Terms
 Average Rate of Change – A quotient that tells, on average, how some quantity changes over time.
 The ratio between the change in the dependent variable and the change in the independent variable.
 Instantaneous Slope – The slope of a line that is tangent to a curve at a point.
 It may be estimated by calculating the average slope between the point and a nearby point on the curve.
 The closer the nearby point, the better the estimation will be.
 In nonlinear functions, the average rate of change and the instantaneous slope are NOT the same thing.
 Vertex – Marks four features of a parabola:
 Its sharpest turn
 Its center
 Its line of symmetry
 The halfway point between its two xintercepts
Notes
Review 
 Review
 Roots of a Polynomial
 The xintercepts
 The places where the graph crosses, or touches, the xaxis
 The xvalues that make y = 0
 Roots are also called Zeros (where the parabola crosses the xaxis)
 To find the zeros of a parabola, set the y equal to 0 and solve for x.
 Use any of the methods learned in this chapter to solve for x.
 Averages
 To find the average of 2 values, add the values together and divide by 2.
 Ex. 4 and 14
 Add 4 + 14: 18
 Divide by 2: 9
 The average of 4 and 14 is 9.

Quadratics’ Graphs are Parabolas 
 When quadratic equations are graphed, they form parabolas.
 The vertex of a parabola is the point at which the direction of the function changes.
 The vertex of a parabola is also either the minimum or the maximum of the function, depending on whether the leading coefficient is positive or negative.
 Minimum: parabola opens upward
 Maximum: parabola opens downward

Finding the Vertex of a Parabola 
 Option 1: Finding the Vertex by Factoring & finding Roots
 1st: find the xintercepts (roots)
 2nd: factor
 3rd: set equal to zero & solve
 4th: list the roots as coordinates (x,0) and (x,0)
 5th: find the average of the xvalues
 6th: substitute the average into the original equation and find the yvalue
 Example
 1st & 2nd: factors into (x – 1)(x – 5) = 0
 3rd: x – 1 = 0 results in x = 1
 3rd: x – 5 = 0 results in x = 5
 4th: The roots are (1,0) and (5,0) since we know the roots are on the xaxis (where the yvalue is equal to zero)
 5th: find the average:
 6th: substitute the xvalue for the vertex into the original equation to find the yvalue:
 Remember the signs of the equation and the signs of the binomial factors (x – #)(x + #) can help find the roots

 Option 2: Finding the Vertex of a parabola with the Quadratic Formula
 Formula for finding the xvalue of the vertex:
* Imaginary numbers ARE complex roots
 Example 1:
 a: 1
 b: 6
 c: 5
 Substitute:
 Solve for the xvalue: 3
 Substitute 3 for x and solve for y (see step 6 above)

Finding the Equation of a Parabola 
 Parabolas are symmetrical curves, meaning that each side of the curve is an exact reflection of the other side.
 The line of symmetry (also called the axis of symmetry) that divides the parabola into two mirror images runs directly through its vertex, as shown in the image below.
 To find the equation of a parabola, set up the binomial factors as the opposite values of the roots (zeros)
 When a is positive, the root is (x – a).
 When a is negative, the root is (x + a).
 Ex. Roots: x = 6 and x = 2 will be the factors: (x – 6) and (x – 2). Use FOIL to find the equation:

Important!
 Click on the animations on Pgs 4, 11
 Practice: Pgs 7, 13, 14, 15, 30, 31, 32
Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=1899