Quadratic Functions and Parabolas
Graphing, analyzing, and solving quadratic relationships
Quadratic Functions and Parabolas
Graphing, analyzing, and solving quadratic relationships
Math - Grade 9-12
- 1
Find the vertex of the quadratic function y = x^2 - 6x + 5.
Use the vertex formula x = -b / 2a or complete the square.
The vertex is (3, -4). Completing the square gives y = (x - 3)^2 - 4, so the vertex is at x = 3 and y = -4. - 2
Determine whether the parabola y = -2x^2 + 8x - 1 opens upward or downward. Then state the axis of symmetry.
The parabola opens downward because the coefficient of x^2 is negative. The axis of symmetry is x = 2 because x = -b / 2a = -8 / (2 times -2) = 2. - 3
Write the quadratic function in standard form that has x-intercepts at x = 1 and x = 5 and passes through the point (2, -3).
Start with factored form using the given x-intercepts.
The function is y = x^2 - 6x + 5. Using the intercepts, the function has the form y = a(x - 1)(x - 5). Substituting (2, -3) gives -3 = a(1)(-3), so a = 1. Expanding gives y = x^2 - 6x + 5. - 4
Solve the equation x^2 + 7x + 12 = 0 by factoring.
The solutions are x = -3 and x = -4. Factoring gives (x + 3)(x + 4) = 0, so each factor can be set equal to zero. - 5
Find the y-intercept of the function y = 3x^2 - 2x + 7.
The y-intercept occurs where x = 0.
The y-intercept is 7, or the point (0, 7). Substituting x = 0 into the function gives y = 7. - 6
A ball is thrown upward, and its height is modeled by h(t) = -16t^2 + 48t + 5. Find the maximum height of the ball.
The maximum height is 41 feet. The time of the vertex is t = -b / 2a = -48 / (2 times -16) = 1.5. Substituting t = 1.5 gives h(1.5) = -16(1.5)^2 + 48(1.5) + 5 = 41. - 7
Convert y = x^2 + 4x - 1 into vertex form.
Complete the square by adding and subtracting the same value.
The vertex form is y = (x + 2)^2 - 5. Completing the square shows x^2 + 4x - 1 = (x^2 + 4x + 4) - 4 - 1 = (x + 2)^2 - 5. - 8
State the domain and range of the function y = (x - 1)^2 + 6.
The domain is all real numbers. The range is y is greater than or equal to 6 because the parabola opens upward and has a minimum value of 6 at its vertex. - 9
Find the zeros of y = x^2 - 9.
Use the difference of squares.
The zeros are x = 3 and x = -3. Setting y = 0 gives x^2 - 9 = 0, which factors as (x - 3)(x + 3) = 0. - 10
A quadratic function has vertex (4, -2) and opens upward. Write one possible equation in vertex form.
One possible equation is y = (x - 4)^2 - 2. This matches the vertex form y = a(x - h)^2 + k with h = 4, k = -2, and a positive value of 1. - 11
Solve x^2 - 4x - 5 = 0 using the quadratic formula.
Substitute carefully into x = [-b plus or minus square root of (b^2 - 4ac)] / 2a.
The solutions are x = 5 and x = -1. Using the quadratic formula with a = 1, b = -4, and c = -5 gives x = [4 plus or minus square root of 36] / 2, which simplifies to x = [4 plus or minus 6] / 2. - 12
For the function y = -x^2 + 6x - 8, find the vertex and the maximum value.
The vertex is (3, 1), and the maximum value is 1. Using x = -b / 2a gives x = -6 / (2 times -1) = 3. Substituting x = 3 gives y = -9 + 18 - 8 = 1. - 13
Graphing question: Describe how the graph of y = (x + 1)^2 - 3 is related to the graph of y = x^2.
Use the form y = (x - h)^2 + k to identify shifts.
The graph is shifted 1 unit left and 3 units down from y = x^2. The shape stays the same because the coefficient of the squared term is still 1. - 14
Find the axis of symmetry and x-intercepts of y = x^2 - 2x - 8.
The axis of symmetry is x = 1. The x-intercepts are x = 4 and x = -2, so the intercept points are (4, 0) and (-2, 0). Factoring gives (x - 4)(x + 2) = 0. - 15
A rectangular garden has a length that is 3 feet more than its width. Its area is 54 square feet. Write and solve a quadratic equation to find the dimensions.
Use area = length times width and reject any negative dimension.
Let the width be w. Then the length is w + 3, and the area equation is w(w + 3) = 54, or w^2 + 3w - 54 = 0. Factoring gives (w + 9)(w - 6) = 0, so w = 6 or w = -9. The negative value is not reasonable, so the width is 6 feet and the length is 9 feet.