# 3.7 – Graphs of Quadratic Functions

## Objectives

• Determine the number of times that a given function crosses the x-axis.
• Find the vertex and x-intercepts 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 x-intercepts

## Notes

Review
• Review
• Roots of a Polynomial
• The x-intercepts
• The places where the graph crosses, or touches, the x-axis
• The x-values that make y = 0
• Roots are also called Zeros (where the parabola crosses the x-axis)
• 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.
• 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 x-intercepts (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 x-values
• 6th: substitute the average into the original equation and find the y-value

• Example
• 1st & 2nd: $y=x^2-6x+5$ 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 x-axis (where the y-value is equal to zero)
• 5th: find the average:

• 6th: substitute the x-value for the vertex into the original equation to find the y-value:

• 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 x-value of the vertex: $- \frac{b}{2a}$

* Imaginary numbers ARE complex roots

• Example 1:  $y=x^2-6x+5$
• a: 1
• b: -6
• c: 5
• Substitute:  $- \frac{-6}{(2)(1)}$
• Solve for the x-value:  3
• Substitute 3 for x and solve for y (see step 6 above)

• Example 2

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: $x^2-8x+12$