Parabolas in Vertex Form

Show your work for each problem.

1.
What is the vertex of ?
1. a)
2. b)
3. c)
4. d)
Answer:
From , we get . The constant gives us . So the vertex is .
2.
Which direction does the parabola open?
1. a)Left
2. b)Downward
3. c)Right
4. d)Upward
Answer: Downward
Since , the parabola opens downward. The vertex is the maximum point.
3.
What is the axis of symmetry for ? (Write as x = ?)
Answer: 7
From , we identify . The axis of symmetry is the vertical line .
4.
What is the value in ?
Answer: -3
, so . The vertex is at .
5.
Is the parabola wider or narrower than ?
1. a)Cannot determine
2. b)Same width
3. c)Wider
4. d)Narrower
Answer: Wider
Since and , the parabola is wider than . Smaller means the parabola grows more slowly.
6.
A parabola has vertex and . What is the -value when ?
Answer: 5
. When : .
7.
What is the vertex of ?
1. a)
2. b)
3. c)
4. d)
Answer:
gives . The constant gives . Vertex: .
8.
Write the equation in vertex form for a parabola with vertex that passes through .
Answer: y = -(x - 1)^2 + 6
1. Write the vertex form template with vertex y = a(x - 1)^2 + 6
2. Substitute the point for and 2 = a(3 - 1)^2 + 6
3. Simplify 4
4. Solve for : -1
9.
The maximum height of a ball is 20 meters at time seconds. If the height follows and , what is ?
Answer: -2
. So , giving . The negative value makes sense since the parabola opens downward (maximum at vertex).
10.
Convert to vertex form.
Answer: y = (x + 2)^2 - 3
1. Group the terms: . What value completes the square? 4
2. Add and subtract 4: . Factor the perfect square. (x + 2)^2
3. Simplify the constants: -3
4. What is the vertex? (-2, -3)
11.
Which equation represents a parabola with vertex at the origin that opens downward?
1. a)
2. b)
3. c)
4. d)
Answer:
has vertex and , so it opens downward. The other options either have wrong vertex or wrong direction.
12.
Convert to vertex form.
Answer: y = 2(x - 3)^2 - 4
1. Factor out the coefficient of from the first two terms y = 2(x^2 - 6x) + 14
2. What value completes the square inside the parentheses? 9
3. Add 9 inside and subtract outside. What is ? -4
4. Write the final vertex form y = 2(x - 3)^2 - 4
13.
A parabola in vertex form has vertex and passes through . Find the value of .
Answer: -1
. So , giving .
14.
Convert to vertex form.
Answer: y = -3(x - 1)^2 + 5
1. Factor out from the first two terms y = -3(x^2 - 2x) + 2
2. What value completes the square for ? 1
3. When we add 1 inside, we subtract outside. What is ? 5
4. Write the final vertex form y = -3(x - 1)^2 + 5
15.
A ball is thrown and its height in meters is . What is the maximum height and when does it occur?
1. a)25 meters at seconds
2. b)2 meters at seconds
3. c)20 meters at seconds
4. d)5 meters at seconds
Answer: 25 meters at seconds
The vertex is . Since , the parabola opens downward, so the vertex is the maximum point. Maximum height is 25 meters at seconds.
16.
For the parabola , at what -values does ? (Enter the smaller value)
Answer: -0.5
. So or . The smaller value is .