Quadratics Cheatsheet¶
1. Standard Form¶
\[
f(x) = ax^2 + bx + c
\]
where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\).
2. Vertex Form¶
\[
f(x) = a(x - h)^2 + k
\]
- Vertex: \((h, k)\)
3. Factored Form¶
\[
f(x) = a(x - r_1)(x - r_2)
\]
- Roots: \(r_1\) and \(r_2\)
4. Quadratic Formula¶
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- Discriminant (\(\Delta\)): \(b^2 - 4ac\)
- \(\Delta > 0\): Two distinct real roots
- \(\Delta = 0\): One real root (repeated)
- \(\Delta < 0\): Two complex roots
5. Axis of Symmetry¶
\[
x = -\frac{b}{2a}
\]
6. Vertex¶
- x-coordinate: \(x = -\frac{b}{2a}\)
- y-coordinate: Substitute \(x\) into \(f(x)\)
7. Y-Intercept¶
\[
f(0) = c
\]
8. Roots and Relationships¶
- Sum of the roots:
\[
r_1 + r_2 = -\frac{b}{a}
\]
- Product of the roots:
\[
r_1 \cdot r_2 = \frac{c}{a}
\]
9. Parabola Shape¶
- Upward Opening: \(a > 0\)
- Downward Opening: \(a < 0\)
10. Focus and Directrix¶
For \(y = a(x - h)^2 + k\):
- Focus: \((h, k + \frac{1}{4a})\)
- Directrix: \(y = k - \frac{1}{4a}\)
For \(x = a(y - k)^2 + h\):
- Focus: \((h + \frac{1}{4a}, k)\)
- Directrix: \(x = h - \frac{1}{4a}\)
11. Width and Stretch¶
- Narrower: Larger \(|a|\)
- Wider: Smaller \(|a|\)
12. Applications¶
- Optimization: Vertex provides maximum or minimum value.
- Projectile Motion: Models height vs. time.
- Quadratic Regression: Fits a quadratic model to data.
- Real-World Problems: Area, revenue, dimensions.
This cheatsheet summarizes key concepts and formulas for quadratic functions and equations, aiding in quick reference and application.