Quadratic Inequalities
Solve x² − 9 > 0.
Write the answer in interval notation.
x ∈ (−∞, −3) and (3, ∞)
Find the intersections of y = x² and y = 4.
(−2, 4) and (2, 4)
The points (0, 3), (1, 6), (2, 9), (3, 12) are given. Is a linear or quadratic model more appropriate?
Linear
Graph f(x) = x² − 6x + 8. What are the x-intercepts?
(2, 0) and (4, 0)
Solve x² − 16 ≤ 0.
x ∈ [−4, 4]
Solve x² − 5x + 6 ≤ 0.
x ∈ [2, 3]
Solve the system y = x² − 2 and y = x + 4.
(−2, 2) and (3, 7)
Use quadratic regression for (0, 2), (1, 5), (2, 10), (3, 17). Find the equation.
f(x) = x² + 2x + 2
Graph f(x) = −x² + 8x − 7. What is the maximum value?
9
Solve y = x² − 1 and y = 3.
(−2, 3) and (2, 3)
Solve 2x² − 8x ≥ 0.
x ∈ (−∞, 0] ∪ [4, ∞)
Solve y = x² − 4x + 1 and y = 1.
(0, 1) and (4, 1)
Festival attendance is modeled by f(x) = −2x² + 40x + 100. Predict attendance on day 8.
292 visitors
A ball's height is h(t) = −16t² + 64t + 5. When does it reach its maximum height?
2 seconds
A ball's height is h(t) = −16t² + 48t + 5. Find its height at 2 seconds.
37 ft
Solve x² + 4x − 12 < 0.
x ∈ (−6, 2)
Two numbers have a difference of 5. The square of the first plus the second is 19. Write a system using x and y.
x − y = 5 and x² + y = 19
A sales model is f(x) = 3.25x² − 26x + 180. Predict sales on day 12.
$336
A company's profit is P(x) = −2x² + 40x − 50. Find the maximum profit.
$150
Two positive integers differ by 4. The square of the first minus twice the second is 38. Write a system and find the numbers.
x − y = 4 & x² − 2y = 38.
The numbers are 6 and 2.
Solve 2x² − 6x − 20 ≥ 0. Explain why there are two intervals.
x ∈ (−∞, −2] & [5, ∞)
The zeros are −2 and 5. The parabola opens upward, so it is nonnegative outside the roots.
Solve algebraically
y = x² − 5x + 2
y = x − 6
(2, −4) and (4, −2)
A store records sales (0, 120), (1, 105), (2, 100), (3, 105), (4, 120). Find the quadratic regression equation and predict day 6.
f(x) = 5x² − 20x + 120; day 6 = $180
Store sales are S(x) = 3x² − 30x + 200. Find the minimum sales amount and when it occurs.
$125 on day 5
A penny falls from a 64-foot building. Its heights at 0, 0.5, 1, and 1.5 seconds are 64, 60, 48, and 28 feet. Find a quadratic regression model and determine when it reaches the ground.
h(t) = −16t² + 64
The penny reaches the ground after 2 seconds.