Notes
Three forms of a quadratic
| Form | Equation | Shows |
|---|---|---|
| Standard | $y = ax^2 + bx + c$ | $y$-intercept $c$; vertex at $\displaystyle x = -\frac{b}{2a}$ |
| Vertex | $y = a(x - h)^2 + k$ | vertex $(h, k)$ |
| Factored | $y = a(x - p)(x - q)$ | $x$-intercepts $p$ and $q$ |
The parabola opens up when $a > 0$ and down when $a < 0$.
Solving
- Factor, take square roots (remember $\pm$), complete the square, or use the quadratic formula $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
- The discriminant $b^2 - 4ac$: positive means two real solutions, zero means one, negative means two nonreal complex solutions.
Complex numbers
- $\displaystyle i = \sqrt{-1}$ and $i^2 = -1$. Complex numbers look like $a + bi$.
- Add by combining like parts; multiply with FOIL and replace $i^2$ with $-1$.
- Divide by multiplying top and bottom by the conjugate $a - bi$.
Quadratic inequalities
Find the zeros, then decide which intervals make the inequality true. For $a > 0$, "$< 0$" is between the zeros and "$> 0$" is outside them.
Topics in this unit
Each topic has its own short notes and practice.
Practice
Add and subtract 9: $(x^2 + 6x + 9) + 2 - 9 = (x + 3)^2 - 7$.
$4 - 1 = 3$ and $-3i - (-5i) = 2i$.
$1 + 4i + 4i^2 = 1 + 4i - 4$.
$(-4)^2 - 4(3)(2) = 16 - 24 = -8 < 0$.
$\displaystyle x = \frac{-4 \pm \sqrt{16 - 52}}{2} = \frac{-4 \pm 6i}{2}$.
$x^2 - 5x \le 0 \Rightarrow x(x - 5) \le 0$, true between the zeros 0 and 5.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A ball is thrown upward from the roof of a building. Its height in feet after $t$ seconds is $h(t) = -16t^2 + 64t + 80$.
(a) Find the maximum height of the ball and when it occurs.
Show the answer and scoring guide
$\displaystyle t = -\frac{64}{2(-16)} = 2$ seconds. $h(2) = -64 + 128 + 80 = 144$ feet.
Check off the points you earned:
(b) When does the ball hit the ground?
Show the answer and scoring guide
$-16t^2 + 64t + 80 = 0 \Rightarrow t^2 - 4t - 5 = 0 \Rightarrow (t - 5)(t + 1) = 0$. Reject $t = -1$, so $t = 5$ seconds.
Check off the points you earned:
(c) For what times is the ball higher than 128 feet?
Show the answer and scoring guide
$-16t^2 + 64t + 80 > 128 \Rightarrow t^2 - 4t + 3 < 0 \Rightarrow (t - 1)(t - 3) < 0$, so $1 < t < 3$ seconds.
Check off the points you earned: