Review of Quadratic Functions and Graphs

Unit 2: Understanding Parabolas and Quadratic Functions

Writing Vertex Form Equations of Parabolas

1. Equation: y = x² + 16x + 71

• Transformations:

  • Complete the square to rewrite in vertex form.

  • The vertex form is given by:
    y = a(x-h)² + k
    where (h, k) is the vertex.

• Vertex Calculation:

  • To complete the square, use the formula: h = -\frac{b}{2a}, here, b = 16 and a = 1.

    • Thus, h = -\frac{16}{2(1)} = -8.

  • Substitute back to find k:
    k = (-8)^2 + 16(-8) + 71 = 71 - 128 + 64 = 7.

  • Vertex: (-8, 7)

• Axis of Symmetry:

  • Line of symmetry: x = -8

2. Equation: y = x² - 2x - 5

• Transformations:

  • Complete the square.

• Vertex Calculation:

  • h = -\frac{-2}{2(1)} = 1.

  • Substitute back:
    k = (1)^2 - 2(1) - 5 = 1 - 2 - 5 = -6.

  • Vertex: (1, -6)

• Axis of Symmetry:

  • Line of symmetry: x = 1

3. Equation: y = -x² - 14x - 59

• Transformations:

  • Complete the square.

• Vertex Calculation:

  • h = -\frac{-14}{2(-1)} = -7.

  • Substitute back:
    k = -(-7)² - 14(-7) - 59 = -49 + 98 - 59 = -10.

  • Vertex: (-7, -10)

• Axis of Symmetry:

  • Line of symmetry: x = -7

4. Equation: y = 2x² + 36x + 170

• Transformations:

  • Complete the square.

• Vertex Calculation:

  • h = -\frac{36}{2(2)} = -9.

  • Substitute back:
    k = 2(-9)² + 36(-9) + 170 = 162 - 324 + 170 = 8.

  • Vertex: (-9, 8)

• Axis of Symmetry:

  • Line of symmetry: x = -9

Sketching the Graph

• For each parabola, visualize the U-shape where:

  • If a > 0, the parabola opens upwards.

  • If a < 0, it opens downwards.

• Locate and plot the vertex and draw the axis of symmetry accordingly.

Factorization

Factor the following polynomials:

1. Polynomial: 2x² - 5x - 3

   • To factor: Look for two numbers that multiply to: 2 * (-3) = -6 and add to: -5.

   • Factors: 2x + 1 and x - 3.

   • Factored form: (2x + 1)(x - 3).

2. Polynomial: 3x² + 10x - 8

   • Find numbers that multiply to: 3 * (-8) = -24 and add to: 10.

   • Factors: 12 and -2.

   • Factored form: (3x - 2)(x + 4).

3. Polynomial: 2n² - 3n - 14

   • To factor: Numbers that multiply to: 2 * (-14) = -28 and add to: -3.

   • Factored form: (2n + 7)(n - 2).

4. Polynomial: 5n² + 2n + 7

   • This polynomial is prime as it has no real roots.

Motion and Height Functions

Jason’s Cliff Jumping Scenario

1. Function: h(t) = -16t² + 16t + 480

   • Maximum Height Time Calculation:

     • Time to max height: t = -\frac{b}{2a} = -\frac{16}{2(-16)} = 0.5 seconds.

   • Maximum Height:

     • Substitute into h(t):
       h(0.5) = -16(0.5)² + 16(0.5) + 480 = 4 + 8 + 480 = 492 feet.

   • Total Time to Hit Water:

     • Find roots of h(t) = 0; set -16t² + 16t + 480 = 0.

     • Use the quadratic formula:
       t = \frac{-b ± \sqrt{b² - 4ac}}{2a} gives the time it takes to hit the water.

     • Roots are calculated as follows.

Grappling Hook Release

2. Function: h(t) = -16t² + 32t + 5

   • Maximum Height Calculation:

     • Time to max height:
       t = -\frac{32}{2(-16)} = 1 second.

     • Substitute t = 1 into h(t):
       h(1) = -16(1)² + 32(1) + 5 = 21 feet.

   • Ledge Height Comparison:

     • Compare height of hook (21 feet) to ledge height (20 feet): Yes, you can reach the ledge.

Solving Quadratic Equations

Solve by Factoring:

1. Equation: x² - 8x + 16 = 0

   • This factors to (x - 4)(x - 4) = 0.

   • Thus, the solution is: x = 4.

Solve by Taking Square Roots:

1. Equation: -8 - 5n² = -88

   • Rearranging gives: 5n² = 80, leading to n² = 16.

   • Solutions: n = ±4.

Solve using the Quadratic Formula:

1. Equation: 4v² + 7v - 7 = 0

   • Using quadratic formula:
     v = \frac{-7 ± \sqrt{(7)² - 4(4)(-7)}}{2(4)}.

   • Thus, calculate the discriminant and continue solving.

2. Equation: 2n² - 18n + 40 = 0

   • Use quadratic formula as above.

3. Equation: 4 - 2a² = -7

   • Rearrange to standard form before applying the quadratic formula.

4. Equation: -8b² - 3b + 22 = 0

   • Apply quadratic formula to solve for b.