Quadratic Functions
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Find the Maximum or Minimum Value
Standard Form:
\[ f(x) = ax^2 + bx + c \]
Determine Max or Min:
• If \( a > 0 \): Parabola opens UP → has a MINIMUM
• If \( a < 0 \): Parabola opens DOWN → has a MAXIMUM
Vertex Formula:
\[ h = -\frac{b}{2a} \quad \text{(x-coordinate of vertex)} \]
\[ k = f(h) = f\left(-\frac{b}{2a}\right) \quad \text{(y-coordinate = max/min value)} \]
Vertex Form:
\[ f(x) = a(x - h)^2 + k \]
The vertex is \( (h, k) \), and \( k \) is the max/min value
Example:
Find the minimum value of \( f(x) = 2x^2 - 8x + 5 \)
Since \( a = 2 > 0 \), parabola opens up → has minimum
\( h = -\frac{-8}{2(2)} = \frac{8}{4} = 2 \)
\( k = f(2) = 2(2)^2 - 8(2) + 5 = 8 - 16 + 5 = -3 \)
Minimum value: -3 at x = 2
2. Characteristics of Quadratic Functions
Key Features:
1. Vertex \( (h, k) \):
Turning point (maximum or minimum)
2. Axis of Symmetry:
\[ x = h = -\frac{b}{2a} \]
Vertical line through vertex that divides parabola into mirror images
3. Y-intercept:
Point where graph crosses y-axis: \( (0, c) \) or \( f(0) = c \)
4. X-intercepts (Zeros/Roots):
Points where graph crosses x-axis; found by solving \( f(x) = 0 \)
5. Domain:
All real numbers: \( (-\infty, \infty) \)
6. Range:
• If \( a > 0 \): \( [k, \infty) \) (minimum value is k)
• If \( a < 0 \): \( (-\infty, k] \) (maximum value is k)
7. Direction of Opening:
• \( a > 0 \): Opens upward (U-shaped)
• \( a < 0 \): Opens downward (∩-shaped)
8. Width:
• \( |a| > 1 \): Narrower than \( y = x^2 \)
• \( 0 < |a| < 1 \): Wider than \( y = x^2 \)
3. Graph a Quadratic Function
Steps to Graph:
Step 1: Find the vertex
Use \( h = -\frac{b}{2a} \) and \( k = f(h) \)
Step 2: Draw the axis of symmetry
Vertical line at \( x = h \)
Step 3: Find the y-intercept
Point \( (0, c) \)
Step 4: Find x-intercepts (if they exist)
Solve \( ax^2 + bx + c = 0 \)
Step 5: Plot additional points
Choose x-values on both sides of the vertex and find corresponding y-values
Step 6: Draw the parabola
Connect points with smooth U-shaped curve
4. Solve Using Square Roots
Method:
Works when the equation can be written as \( (x - p)^2 = q \) or \( x^2 = k \)
Square Root Property:
\[ \text{If } x^2 = k, \text{ then } x = \pm\sqrt{k} \]
Example:
Solve: \( 3(x - 2)^2 = 27 \)
\( (x - 2)^2 = 9 \)
\( x - 2 = \pm 3 \)
\( x = 2 + 3 = 5 \) or \( x = 2 - 3 = -1 \)
Solutions: x = 5, x = -1
5. Solve by Factoring
Method:
Steps:
1. Write equation in standard form: \( ax^2 + bx + c = 0 \)
2. Factor the left side
3. Use Zero Product Property: If \( AB = 0 \), then \( A = 0 \) or \( B = 0 \)
4. Solve each equation
Example:
Solve: \( x^2 + 5x - 14 = 0 \)
Factor: \( (x + 7)(x - 2) = 0 \)
Set each factor equal to zero:
\( x + 7 = 0 \) → \( x = -7 \)
\( x - 2 = 0 \) → \( x = 2 \)
Solutions: x = -7, x = 2
6. Solve by Completing the Square
Method:
Steps:
1. Move constant to right side: \( x^2 + bx = -c \)
2. Add \( \left(\frac{b}{2}\right)^2 \) to both sides
3. Factor left side as perfect square: \( \left(x + \frac{b}{2}\right)^2 = \text{result} \)
4. Use square root property to solve
Key Formula:
\[ \text{To complete the square, add } \left(\frac{b}{2}\right)^2 \]
Example:
Solve: \( x^2 + 6x - 7 = 0 \)
Move constant: \( x^2 + 6x = 7 \)
Add \( \left(\frac{6}{2}\right)^2 = 9 \) to both sides:
\( x^2 + 6x + 9 = 7 + 9 \)
\( (x + 3)^2 = 16 \)
\( x + 3 = \pm 4 \)
\( x = -3 + 4 = 1 \) or \( x = -3 - 4 = -7 \)
Solutions: x = 1, x = -7
7. Solve Using the Quadratic Formula
The Quadratic Formula:
For \( ax^2 + bx + c = 0 \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Steps:
1. Identify a, b, and c from standard form
2. Substitute into the formula
3. Simplify under the square root first
4. Compute both solutions (+ and −)
Example:
Solve: \( 2x^2 + 3x - 5 = 0 \)
Identify: \( a = 2, b = 3, c = -5 \)
\( x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)} \)
\( x = \frac{-3 \pm \sqrt{9 + 40}}{4} \)
\( x = \frac{-3 \pm \sqrt{49}}{4} = \frac{-3 \pm 7}{4} \)
\( x = \frac{-3 + 7}{4} = 1 \) or \( x = \frac{-3 - 7}{4} = -\frac{5}{2} \)
Solutions: x = 1, x = -2.5
8. Using the Discriminant
Discriminant Formula:
\[ \Delta = b^2 - 4ac \]
The discriminant tells us the nature and number of solutions
Discriminant Cases:
| Discriminant Value | Nature of Roots | Number of Solutions |
|---|---|---|
| \( \Delta > 0 \) | Two distinct real roots | 2 solutions |
| \( \Delta = 0 \) | One repeated real root (double root) | 1 solution |
| \( \Delta < 0 \) | Two complex (imaginary) roots | 0 real solutions |
Example:
Determine the nature of roots for \( x^2 - 4x + 4 = 0 \)
\( a = 1, b = -4, c = 4 \)
\( \Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0 \)
Since Δ = 0, there is ONE repeated real root (x = 2)
9. Solve Quadratic Equations: Word Problems
Common Problem Types:
1. Area Problems
Rectangles, triangles where dimensions form quadratic equations
2. Projectile Motion
Height formula: \( h(t) = -16t^2 + v_0t + h_0 \) (feet) or \( h(t) = -4.9t^2 + v_0t + h_0 \) (meters)
3. Number Problems
Consecutive integers, product relationships
4. Business/Revenue Problems
Profit = Revenue - Cost, where equations are quadratic
Problem-Solving Steps:
1. Read carefully and identify what you're looking for
2. Define variables
3. Write equation based on the problem
4. Solve the equation
5. Check if solutions make sense in context
6. Answer the question asked
10. Quick Reference Summary
Key Formulas:
Standard Form: \( f(x) = ax^2 + bx + c \)
Vertex Form: \( f(x) = a(x - h)^2 + k \)
Vertex: \( \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) \)
Axis of Symmetry: \( x = -\frac{b}{2a} \)
Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Discriminant: \( \Delta = b^2 - 4ac \)
Completing the Square: Add \( \left(\frac{b}{2}\right)^2 \)
📚 Study Tips
✓ Always check if a > 0 (opens up) or a < 0 (opens down) first
✓ Use discriminant to quickly determine nature of solutions
✓ Quadratic formula works for ALL quadratic equations
✓ Factoring is fastest when it works easily
✓ Vertex form is best for finding max/min values quickly