Question

kadeesha throws a ball up in the air. the graph below shows the height of the ball h in feet after t seconds.

Explanation:

Step1: Identify the graph type

The graph of the ball's height over time is a parabola, so it represents a quadratic function. The general form of a quadratic function in vertex form is \( h(t) = a(t - h)^2 + k \), where \((h, k)\) is the vertex. Here, the vertex is \((0.75, 9)\), so \( h(t) = a(t - 0.75)^2 + 9 \).

Step2: Use the root to find \( a \)

We know the graph passes through \((1.5, 0)\). Substitute \( t = 1.5 \) and \( h = 0 \) into the equation:

$$ 0 = a(1.5 - 0.75)^2 + 9 $$

$$ 0 = a(0.75)^2 + 9 $$

$$ 0 = 0.5625a + 9 $$

Subtract 9 from both sides:

$$ -9 = 0.5625a $$

Divide both sides by 0.5625:

$$ a = \frac{-9}{0.5625} = -16 $$

Step3: Write the function

Now that we have \( a = -16 \), the function is \( h(t) = -16(t - 0.75)^2 + 9 \). We can also expand this to standard form, but for most questions about this graph (like finding maximum height, time to hit the ground, etc.), the vertex form is useful. The maximum height is the \( k \)-value of the vertex, which is 9 feet, and the time to hit the ground is when \( h(t) = 0 \), which we already used \( t = 1.5 \) seconds.

Answer:

If the question was about maximum height, the answer is 9 feet. If about time to hit the ground, 1.5 seconds. (Assuming a common question about this projectile motion graph, the maximum height is 9 feet at \( t = 0.75 \) seconds, and it hits the ground at \( t = 1.5 \) seconds.)