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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.
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

Alg2A 3.07 - Maximum Vertex Alg2A 3.07 - Minimum Vertex

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:

Alg2A 3.07 - Vertex y

  • Remember the signs of the equation and the signs of the binomial factors (x – #)(x + #) can help find the roots

Alg2A 3.07 - Ex Sign Chart

  • Option 2:  Finding the Vertex of a parabola with the Quadratic Formula
    • Formula for finding the x-value of the vertex: - \frac{b}{2a}

Alg2A 3.07 - Rules for Quadratics w Vertex

Alg2A 3.07 - Roots and Discriminants

* 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

Alg2A 3.07 - Vertex through Quad Eq

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.

Alg2A 3.07 - Axis of Symmetry

  • 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

Alg2A 3.07 - Writing Equation with Roots

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