Examples with Quadratic Functions

Example

Graph $LaTeX: f\left(x\right)=x^2-2x-8.$ $f\left(x\right)=x^2-2x-8.$

Solution

Try this example, and then watch this video for the solution.

Examples

Complete the square to convert the following quadratic functions, given in the form $LaTeX: f\left(x\right)=ax^2+bx+c\:$ $f\left(x\right)=ax^2+bx+c\:$ to equivalent equations for the same functions in the form $LaTeX: f\left(x\right)=a\left(x-h\right)^2+k.$ $f\left(x\right)=a\left(x-h\right)^2+k.$

$y=x^2+4x+3$
$LaTeX: f\left(x\right)=3x^2+12x-1$ $f\left(x\right)=3x^2+12x-1$
$LaTeX: y=\frac{1}{4}x^2-2x+5$ $y=\frac{1}{4}x^2-2x+5$

Solutions

Examples

A gardener has 140 feet of fencing to put around a rectangular vegetable garden. The function $LaTeX: A\left(w\right)=70w-w^2\:$ $A\left(w\right)=70w-w^2\:$ gives the garden's area in square feet for any width in feet. Does the gardener have enough fencing for the area of the garden to be 1300 square feet?
If you drop a water balloon, how long does it take to drop 4 feet? The equation $LaTeX: d\left(t\right)=16t^2\:$ $d\left(t\right)=16t^2\:$ describes the distance the balloon has fallen after t seconds.
The rooftop of a 5 story building is 50 feet above the ground. How long does it take a water balloon dropped from the rooftop to pass by the third story window at 24 feet? How long does it take the water balloon to hit the ground? Use the formula $LaTeX: h\left(t\right)=h_0-16t^2\:$ $h\left(t\right)=h_0-16t^2\:$ to find the height of the balloon, with $LaTeX: h_0\:$ $h_0\:$ representing the initial height, and $LaTeX: t\:$ $t\:$ representing the time after being dropped.
Matt and his friends are enjoying the afternoon at a baseball game. A batter hits a towering home run, and Matt shouts "Wow, that must have been 110 feet high!" The ball was 4 feet off the ground when the batter hit, and the ball comes off traveling vertically at 80 ft/sec.
1. Model the ball's height h (in feet) at time t (n seconds) using the projectile motion model, $LaTeX: h\left(t\right)=-16t^2+v_0t+h_0\:$ $h\left(t\right)=-16t^2+v_0t+h_0\:$ where $LaTeX: v_0\:$ $v_0\:$ is the projectile's initial vertical velocity (in ft/sec) and $LaTeX: h_0\:$ $h_0\:$ is the projectile's initial height (in feet).
2. Use the model to determine the maximum height the ball reaches. What part of the function are we trying to find?
3. Determine how long it took the ball to reach the maximum height. What part of the function are we trying to find?
4. Determine how long it took the ball to hit the ground.

Solutions

Watch the following video solutions once you have attempted the problems above.