MAT140 Section 4.3
Worksheet on Quadratic Functions
Example 1. Consider the quadratic function f (x) = x2 + 8x + 15. y
1. What is the domain of f ? 14 2. Is the graph of f concave up or down? 10 3. What is the vertex (h, k) of f ? 6 4. What is the axis of symmetry of f ? 2 −8
5. Find all x- and y-intercepts of f .
−6
−4
−2
2 x
−2
6. Use your answers to draw a sketch of the graph of f . 7. What is the range of f written in interval notation?
8. Use interval notation to describe the values of x where f is increasing.
Example 2. Consider the quadratic function f (x) = 2x2 − 9x + 10. y
1. What is the domain of f ? 14 2. Is the graph of f concave up or down? 10 3. What is the vertex (h, k) of f ? 6 4. What is the axis of symmetry of f ? 2 −1
5. Find all x- and y-intercepts of f .
−2
1
6. Use your answers to draw a sketch of the graph of f . 7. What is the range of f written in interval notation?
8. Use interval notation to describe the values of x where f is increasing.
1
2
3
4
5
x
Problem 3. Consider the quadratic function f (x) = 4x2 − 4x − 3. 1. What is the domain of f ?
y 6
2. Is the graph of f concave up or down?
4
3. What is the vertex (h, k) of f ?
2 −2
4. What is the axis of symmetry of f ?
−1
1
2
3 x
−2 5. Find all x- and y-intercepts of f .
−4
6. Use your answers to draw a sketch of the graph of f . 7. What is the range of f written in interval notation?
8. Use interval notation to describe the values of x where f is increasing.
Problem 4. Find a quadratic function that has a local minimum at the point (2, 1) and a y-intercept at the point (0, 5). What are the x-intercepts of this function?
Problem 5. Find a quadratic function that has x-intercepts at the points (2, 0) and (−1, 0) and has y-intercept at the point (0, −1). What point is the vertex of the graph of this function?
2
Example 1 Answers 1. Domain: (−∞, ∞) 2. Concave up 3. Vertex (h, k) = (−4, −1) is a local minimum 4. Symmetric about x = −4 5. x-intercepts: (−5, 0) and (−3, 0); y-intercept: (0, 15) 7. Range: [−1, ∞) 8. f (x) is increasing on the interval (−4, ∞)
Example 2 Answers 1. Domain: (−∞, ∞) 2. Concave up 3. Vertex (h, k) = (9/4, −1/8) is a local minimum 4. Symmetric about x = 9/4 5. x-intercepts: (2, 0) and (5/2, 0); y-intercept: (0, 10) 7. Range: [−1/8, ∞) 8. f (x) is increasing on the interval (9/4, ∞)
Problem 3 Answers 1. Domain: (−∞, ∞) 2. Concave up 3. Vertex (h, k) = (1/2, −4) is a local minimum 4. Symmetric about x = 1/2 5. x-intercepts: (1/2, 0) and (3/2, 0); y-intercept: (0, −3) 7. Range: [−4, ∞) 8. f is increasing on the interval (1/2, ∞) Problem 4 Answer: f (x) = (x − 2)2 + 1 1 Problem 5 Answer: f (x) = (x − 2)(x + 1) 2
3
