2. the height of a baseball hit in the air is given by the graphed parabola below. fill in the blank, and then circle the feature the sentence is referring to.
a. the initial height of the ball when it was hit by the bat was ____ feet. (vertex / axis of symmetry / zero / or y-intercept ?)
b. the maximum height of the ball was approximately ____ feet. (vertex / axis of symmetry / zero / or y-intercept ?)
c. after ____ seconds the ball was at its maximum height. (vertex / axis of symmetry / zero / or y-intercept ?)
d. after approximately ____ seconds the ball hit the ground. (vertex / axis of symmetry / zero / or y-intercept ?)
graph: height (feet) vs time (seconds), parabola starting at (0, ~5), peaking around (2.5, 70), ending at (4, 0)
3. identify the features of the following tabular quadratic functions.
table 1: x: -2, -1, 0, 1, 2, 3, 4, 5; f(x): 0, 5, 8, 9, 8, 5, 0, 7
vertex:
axis of symmetry:
zeros (roots/x-intercepts):
y-intercept:
table 2: x: -6, -5, -4, -3, -2, -1, 0, 1; g(x): 14, 0, -10, -16, -18, -16, -10, 0
vertex:
axis of symmetry:
zeros (roots/x-intercepts):
y-intercept:
table 3: x: -1, 0, 1, 2, 3, 4, 5, 6; h(x): -12, -5, 0, 3, 4, 3, 0, -5
vertex:
axis of symmetry:
zeros (roots/x-intercepts):
y-intercept:

Question

Accepted Answer

### Problem 2
#### Part a
# Explanation:
## Step1: Identify initial height
Initial height is at $ t = 0 $ (time when hit), which is the $ y $-intercept. From the graph, at $ x = 0 $ (time = 0), $ y \approx 5 $ feet.
## Step2: Determine the feature
The $ y $-intercept gives the value when $ x = 0 $, so the feature is $ y $-intercept.
# Answer:
5; $ y $-intercept (circle $ y $-intercept)

#### Part b
# Explanation:
## Step1: Find maximum height
Maximum height is the peak of the parabola, which is the vertex's $ y $-coordinate. From the graph, the vertex's $ y $-value is approximately 70 feet.
## Step2: Determine the feature
The vertex is the highest point (for a downward - opening parabola), so the feature is vertex.
# Answer:
70; Vertex (circle Vertex)

#### Part c
# Explanation:
## Step1: Find time at max height
The time at maximum height is the $ x $-coordinate of the vertex. From the graph, the vertex is at $ x\approx2.25 $ (or around 2 - 2.5, more precisely around 2.25) seconds.
## Step2: Determine the feature
The $ x $-coordinate of the vertex gives the time of maximum height, and the vertex is the point of maximum, so the feature is vertex.
# Answer:
2.25 (or 2 - 2.5, e.g., 2.25); Vertex (circle Vertex)

#### Part d
# Explanation:
## Step1: Find time when ball hits ground
The ball hits the ground when height $ y = 0 $, which is the $ x $-intercept (zero) of the parabola. From the graph, $ y = 0 $ at $ x\approx4 $ seconds.
## Step2: Determine the feature
The zeros (x - intercepts) are the points where $ y = 0 $, so the feature is Zero.
# Answer:
4; Zero (circle Zero)

### Problem 3
#### First Table ( $ f(x) $)
# Explanation:
## Step1: Find Vertex
The vertex is the point where the function changes direction. For a quadratic function, the vertex is at the $ x $-value where the function is symmetric. Looking at the table, $ f(x) $ is symmetric around $ x = 1 $ (since $ f(- 2)=f(4) = 0 $, $ f(-1)=f(3)=5 $, $ f(0)=f(2)=8 $ and $ f(1) = 9 $ is the maximum). So the vertex is $ (1,9) $.
## Step2: Find Axis of Symmetry
The axis of symmetry is the vertical line $ x = h $ where $ (h,k) $ is the vertex. So axis of symmetry is $ x = 1 $.
## Step3: Find Zeros (roots/x - intercepts)
Zeros are the $ x $-values where $ f(x)=0 $. From the table, $ f(-2)=0 $ and $ f(4)=0 $, so zeros are $ x=-2 $ and $ x = 4 $.
## Step4: Find y - intercept
The $ y $-intercept is the value of $ f(x) $ when $ x = 0 $. From the table, $ f(0)=8 $, so $ y $-intercept is 8.
# Answer:
Vertex: $ (1,9) $
Axis of Symmetry: $ x = 1 $
Zeros (roots/x - intercepts): $ x=-2, x = 4 $
y - intercept: 8

#### Second Table ( $ g(x) $)
# Explanation:
## Step1: Find Vertex
The function $ g(x) $ is symmetric. Looking at the table, $ g(-5)=g(1)=0 $, $ g(-4)=g(0)=- 10 $, $ g(-3)=g(-1)=-16 $ and $ g(-2)=-18 $ is the minimum. So the vertex is $ (-2,-18) $.
## Step2: Find Axis of Symmetry
The axis of symmetry is the vertical line $ x = h $ where $ (h,k) $ is the vertex. So axis of symmetry is $ x=-2 $.
## Step3: Find Zeros (roots/x - intercepts)
Zeros are the $ x $-values where $ g(x)=0 $. From the table, $ g(-5)=0 $ and $ g(1)=0 $, so zeros are $ x=-5 $ and $ x = 1 $.
## Step4: Find y - intercept
The $ y $-intercept is the value of $ g(x) $ when $ x = 0 $. From the table, $ g(0)=-10 $, so $ y $-intercept is - 10.
# Answer:
Vertex: $ (-2,-18) $
Axis of Symmetry: $ x=-2 $
Zeros (roots/x - intercepts): $ x=-5, x = 1 $
y - intercept: - 10

#### Third Table ( $ h(x) $)
# Explanation:
## Step1: Find Vertex
The function $ h(x) $ is symmetric. Looking at the table, $ h(-1)=h(6)=-12 $, $ h(0)=h(5)=-5 $, $ h(1)=h(4)=0 $ and $ h(2)=h(3)=3 $ and $ h(3)=4 $ is the maximum (wait, $ h(3)=4 $ and $ h(4)=3 $, $ h(2)=3 $, so the vertex is at $ x = 3 $, $ h(3)=4 $). So the vertex is $ (3,4) $.
## Step2: Find Axis of Symmetry
The axis of symmetry is the vertical line $ x = h $ where $ (h,k) $ is the vertex. So axis of symmetry is $ x = 3 $.
## Step3: Find Zeros (roots/x - intercepts)
Zeros are the $ x $-values where $ h(x)=0 $. From the table, $ h(1)=0 $ and $ h(5)=0 $, so zeros are $ x = 1 $ and $ x = 5 $.
## Step4: Find y - intercept
The $ y $-intercept is the value of $ h(x) $ when $ x = 0 $. From the table, $ h(0)=-5 $, so $ y $-intercept is - 5.
# Answer:
Vertex: $ (3,4) $
Axis of Symmetry: $ x = 3 $
Zeros (roots/x - intercepts): $ x = 1, x = 5 $
y - intercept: - 5

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QUESTION IMAGE

Question

Explanation:

Problem 2

Part a

Step1: Identify initial height

Step2: Determine the feature

Step1: Find maximum height

Step2: Determine the feature

Step1: Find time at max height

Step2: Determine the feature

Answer:

Part b