Graphing Quadratic Functions (Parabolas)

Show your work for each problem.

1.
For the function , what is the value of coefficient ?
1. a)
2. b)
3. c)
4. d)
Answer:
In the standard form , the coefficient is the number in front of . Here, , , and .
2.
A parabola with will:
1. a)Open upward and have a maximum
2. b)Open downward and have a minimum
3. c)Open downward and have a maximum
4. d)Open upward and have a minimum
Answer: Open downward and have a maximum
Since , the parabola opens downward. A downward-opening parabola has a highest point (maximum) at its vertex.
3.
What is the y-intercept of ?
Answer: 7
The y-intercept is the value of in , or you can calculate . The y-intercept is at the point .
4.
Find the axis of symmetry for . Give your answer as
Answer: -3
Using . The axis of symmetry is the vertical line .
5.
The discriminant of is:
1. a) (two real roots)
2. b) (one real root)
3. c) (two real roots)
4. d) (no real roots)
Answer: (no real roots)
. Since , the parabola has no real x-intercepts (it doesn't cross the x-axis).
6.
Find the vertex of .
Answer: (4, -4)
1. What are the values of , , and ? a=1, b=-8, c=12
2. Calculate the x-coordinate of the vertex using 4
3. Find to get the y-coordinate -4
4. Write the vertex as a coordinate pair (4, -4)
7.
Find the x-intercepts of .
Answer: x = 2, x = 3
1. Set to find x-intercepts. What equation do you need to solve? x^2 - 5x + 6 = 0
2. Factor the quadratic. What two numbers multiply to 6 and add to -5? -2, -3
3. Write the factored form (x-2)(x-3) = 0
4. Solve for x by setting each factor to zero x = 2, x = 3
8.
The parabola has a maximum or minimum value? What is that value?
Answer: 5
Since , the parabola opens downward and has a maximum. The x-coordinate of the vertex is . The maximum value is .
9.
Which parabola is narrower (steeper)?
1. a)
2. b)
3. c)
4. d)
Answer:
The coefficient determines the width of the parabola. Larger means narrower (steeper). Since , is the narrowest.
10.
Graph by finding the vertex, y-intercept, and x-intercepts.
Answer: Vertex: (-1, -4), y-int: (0, -3), x-int: (-3, 0) and (1, 0)
1. Find the axis of symmetry -1
2. Find the y-coordinate of the vertex by calculating -4
3. Find the y-intercept -3
4. Factor to find the x-intercepts (x+3)(x-1)
5. Solve for the x-intercepts x = -3, x = 1
11.
A ball is thrown upward with height meters after seconds. What is the maximum height reached?
Answer: 22
Time at max height: seconds. Max height: meters.
12.
Analyze : find direction, vertex, axis of symmetry, y-intercept, and x-intercepts.
Answer: Opens down, vertex (2, 9), axis x = 2, y-int (0, 5), x-int (-1, 0) and (5, 0)
1. Does the parabola open up or down? Why? down, a = -1 < 0
2. Find the axis of symmetry x = 2
3. Find the vertex (x-coordinate is 2) (2, 9)
4. Find the y-intercept (0, 5)
5. Factor and find the x-intercepts x = -1, x = 5
13.
A company's revenue is modeled by where is the price in euros. What price maximizes revenue?
1. a)25 euros
2. b)20 euros
3. c)10 euros
4. d)50 euros
Answer: 10 euros
The maximum occurs at euros. At this price, revenue is euros.
14.
Use the discriminant to determine how many x-intercepts has, then explain what this means for the graph.
Answer: 0 x-intercepts, parabola is entirely above x-axis
1. Identify , , and a=2, b=-3, c=5
2. Calculate the discriminant -31
3. Is positive, zero, or negative? negative
4. How many x-intercepts does the parabola have? 0
5. Since , does the parabola open up or down? Where is it relative to the x-axis? opens up, entirely above x-axis
15.
For , what is the minimum value of the function?
Answer: 4
With and , the axis of symmetry is . The minimum is . The vertex is .
16.
Find all key features of : vertex, axis of symmetry, y-intercept, and x-intercepts.
Answer: Vertex: (3, -1), axis: x = 3, y-int: 8, x-int: 2 and 4
1. Find the axis of symmetry x = 3
2. Calculate the vertex's y-coordinate: -1
3. Write the vertex (3, -1)
4. Find the y-intercept 8
5. Factor and find x-intercepts x = 2, x = 4