Example 1
Completion requirements
| Example 1 |
(Video in Development)
Huston is practicing his free throw shot for basketball. The free throw line is \(15\) ft from the centre of the hoop. Huston lets go of the ball at a height of \(7\) ft. On his first throw, the ball follows a quadratic function trajectory, and hits a maximum height of \(16\) ft, at a point \(6.5\) ft from the net. The net is \(10\) ft above the ground. The hoop is \(1.5\) ft in diameter. Does Huston get the ball in the net?
Step 1: Draw a diagram.
Step 2: Write down what you know.
The vertex is at \((6.5, 16)\), therefore \(p = 6.5\) and \(q = 16\).
Step 3: Analyze how to solve the problem.
To solve the problem, determine the equation of the quadratic function. Then, solve for the \(y\)-intercept. If it is equal to (or very close to) \(10\), then Huston has likely made a basket.
Step 4: Solve the problem.
Write an initial equation in vertex form, using \(p = 6.5\) and \(q = 16\).
\(\begin{array}{l}
f\left( x \right) = a\left( {x - p} \right)^2 + q \\
f\left( x \right) = a\left( {x - 6.5} \right)^2 + 16 \\
\end{array}\)
Using the ball’s starting point, \((15, 7)\), substitute \(x = 15\) and \(f(x) = 7\) into the equation, and solve for \(a\).
\(\begin{align}
f\left( x \right) &= a\left( {x - 6.5} \right)^2 + 16 \\
7 &= a\left( {15 - 6.5} \right)^2 + 16 \\
- 9 &= \left( {8.5} \right)^2 a \\
- 9 &= 72.25a \\
- 0.124... &= a \\
\end{align}\)
The equation of the function is
\(f\left( x \right) = - 0.124...\left( {x - 6.5} \right)^2 + 16\)
Step 5: Stop and ask if this makes sense.
Should \(a\) be negative? Yes, because the graph of the quadratic function opens downward.
Step 6: Determine if Huston has made a basket.
Start by solving for the \(y\)-intercept by substituting \(x = 0\) into the equation.
\(\begin{array}{l}
f\left( x \right) = - 0.124...\left( {x - 6.5} \right)^2 + 16 \\
f\left( 0 \right) = - 0.124...\left( {0 - 6.5} \right)^2 + 16 \\
f\left( 0 \right) = 10.737... \\
\end{array}\)
This is far above the required \(10\) ft. Huston will have hit the backboard, but may not have made a basket, depending upon the trajectory of the ball after hitting the backboard.
Place the basket on the \(y\)-axis, the vertex at \((6.5, 16)\), and the starting point of Huston’s ball at \((15, 7)\).
The question is asking whether the curve will cross the line \(y = 10\) between \(x = 0\) and \(x = 1.5\).
The question is asking whether the curve will cross the line \(y = 10\) between \(x = 0\) and \(x = 1.5\).
Step 2: Write down what you know.
The vertex is at \((6.5, 16)\), therefore \(p = 6.5\) and \(q = 16\).
Step 3: Analyze how to solve the problem.
To solve the problem, determine the equation of the quadratic function. Then, solve for the \(y\)-intercept. If it is equal to (or very close to) \(10\), then Huston has likely made a basket.
Step 4: Solve the problem.
Write an initial equation in vertex form, using \(p = 6.5\) and \(q = 16\).
\(\begin{array}{l}
f\left( x \right) = a\left( {x - p} \right)^2 + q \\
f\left( x \right) = a\left( {x - 6.5} \right)^2 + 16 \\
\end{array}\)
Using the ball’s starting point, \((15, 7)\), substitute \(x = 15\) and \(f(x) = 7\) into the equation, and solve for \(a\).
\(\begin{align}
f\left( x \right) &= a\left( {x - 6.5} \right)^2 + 16 \\
7 &= a\left( {15 - 6.5} \right)^2 + 16 \\
- 9 &= \left( {8.5} \right)^2 a \\
- 9 &= 72.25a \\
- 0.124... &= a \\
\end{align}\)
The equation of the function is
\(f\left( x \right) = - 0.124...\left( {x - 6.5} \right)^2 + 16\)
Step 5: Stop and ask if this makes sense.
Should \(a\) be negative? Yes, because the graph of the quadratic function opens downward.
Step 6: Determine if Huston has made a basket.
Start by solving for the \(y\)-intercept by substituting \(x = 0\) into the equation.
\(\begin{array}{l}
f\left( x \right) = - 0.124...\left( {x - 6.5} \right)^2 + 16 \\
f\left( 0 \right) = - 0.124...\left( {0 - 6.5} \right)^2 + 16 \\
f\left( 0 \right) = 10.737... \\
\end{array}\)
This is far above the required \(10\) ft. Huston will have hit the backboard, but may not have made a basket, depending upon the trajectory of the ball after hitting the backboard.