Question

1. 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 ?)

Explanation:

Part a

Step1: Identify initial height

The initial height is when time \( t = 0 \), which is the \( y \)-intercept. From the graph, at \( t = 0 \), the height is 5 feet.

Step2: Determine the feature

The \( y \)-intercept represents the value when \( x = 0 \) (here \( x \) is time \( t \)), so the feature is \( y \)-intercept.

Step1: Identify maximum height

The maximum height of a parabola is at the vertex. From the graph, the vertex (peak) has a height of approximately 65 - 70, let's say around 65 - 70, but looking at the graph, the peak is around 65 - 70, more precisely, let's assume from the graph it's about 65 - 70, but likely around 65 - 70, but maybe 65 - 70, but let's check the graph. The \( y \)-axis at the peak is around 65 - 70, so approximately 65 - 70, but maybe 65 - 70, but let's say 65 - 70, but more accurately, the vertex is the highest point, so the maximum height is at the vertex.

Step2: Determine the feature

The vertex of a parabola is the maximum (for downward opening) point, so the feature is vertex.

Step1: Identify time at max height

The time when the ball is at maximum height is the \( x \)-coordinate of the vertex. From the graph, the vertex is at \( t \approx 2 \) seconds (looking at the time axis, the peak is around \( t = 2 \)).

Step2: Determine the feature

The axis of symmetry passes through the vertex, and the \( x \)-coordinate of the vertex is on the axis of symmetry. Wait, no: the time at maximum height is the \( x \)-coordinate of the vertex. Wait, the vertex is a point \((h, k)\) where \( h \) is the time and \( k \) is the height. So the time at maximum height is the \( x \)-coordinate of the vertex, and the feature related to the time of maximum height is the vertex (since the vertex gives the time and height of maximum). Wait, or axis of symmetry? Wait, the axis of symmetry is the vertical line \( x = h \), where \( h \) is the time of maximum height. Wait, the question is "After ____ seconds the ball was at its maximum height". The time is the \( x \)-coordinate of the vertex, and the feature: the vertex is the point, but the axis of symmetry is the line. Wait, no: the vertex is \((h, k)\), so \( h \) is the time, \( k \) is the height. So the time at maximum height is the \( x \)-coordinate of the vertex, so the feature is vertex? Wait, no, the axis of symmetry is \( x = h \), so the time is \( h \), which is on the axis of symmetry. Wait, maybe the question is considering that the time at maximum height is the \( x \)-coordinate of the vertex, so the feature is vertex. Wait, looking at the graph, the vertex is at \( t \approx 2 \) seconds. So the time is around 2 seconds, and the feature is vertex (since the vertex gives the time of maximum height).

Step2: Determine the feature

The vertex's \( x \)-coordinate is the time of maximum height, so the feature is vertex.

Answer:

5; \( y \)-intercept (circle \( y \)-intercept)

Part b