Basic Math

Quadratic functions | Ninth Grade

Quadratic Functions - Ninth Grade Math

Introduction to Quadratic Functions

Quadratic Function: A polynomial function of degree 2
Graph: A U-shaped curve called a parabola
General Form: $f(x) = ax^2 + bx + c$ where $a \neq 0$
Parent Function: $f(x) = x^2$
Three Forms of Quadratic Functions:

1. Standard Form:
$$f(x) = ax^2 + bx + c$$
where $a$, $b$, $c$ are constants and $a \neq 0$

2. Vertex Form:
$$f(x) = a(x - h)^2 + k$$
where $(h, k)$ is the vertex

3. Factored/Intercept Form:
$$f(x) = a(x - p)(x - q)$$
where $p$ and $q$ are x-intercepts (zeros/roots)

1-2. Characteristics of Quadratic Functions

Key Features of Parabolas:
Vertex: The highest or lowest point
Axis of Symmetry: Vertical line through vertex
Y-intercept: Where graph crosses y-axis
X-intercepts (Zeros/Roots): Where graph crosses x-axis
Direction: Opens up or down
Width: Narrow or wide

Direction of Opening

Coefficient $a$ determines direction:

If $a > 0$: Parabola opens UPWARD (U-shape)
  Vertex is MINIMUM point

If $a < 0$: Parabola opens DOWNWARD (∩-shape)
  Vertex is MAXIMUM point

If $|a| > 1$: Parabola is NARROW (vertical stretch)
If $0 < |a| < 1$: Parabola is WIDE (vertical compression)

Vertex

Finding Vertex from Standard Form $f(x) = ax^2 + bx + c$:

x-coordinate of vertex (h):
$$h = -\frac{b}{2a}$$

y-coordinate of vertex (k):
$$k = f(h) = f\left(-\frac{b}{2a}\right)$$
Substitute $h$ back into function

Or use formula:
$$k = c - \frac{b^2}{4a} = \frac{4ac - b^2}{4a}$$

Vertex: $(h, k)$
Example 1: Find vertex of $f(x) = 2x^2 - 8x + 3$

Given: $a = 2$, $b = -8$, $c = 3$

x-coordinate:
$h = -\frac{-8}{2(2)} = \frac{8}{4} = 2$

y-coordinate:
$k = f(2) = 2(2)^2 - 8(2) + 3$
$k = 8 - 16 + 3 = -5$

Vertex: $(2, -5)$

Axis of Symmetry

Axis of Symmetry Formula:

From Standard Form:
$$x = -\frac{b}{2a}$$

From Vertex Form:
$$x = h$$

This is a vertical line passing through the vertex
Example 2: Find axis of symmetry for $f(x) = -x^2 + 6x - 5$

$a = -1$, $b = 6$
$x = -\frac{6}{2(-1)} = -\frac{6}{-2} = 3$

Axis of symmetry: $x = 3$

Intercepts

Y-intercept:
Set $x = 0$ and solve for $y$
For $f(x) = ax^2 + bx + c$, y-intercept is $(0, c)$

X-intercepts (zeros/roots):
Set $y = 0$ and solve $ax^2 + bx + c = 0$
• Factor if possible
• Use quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Discriminant: $\Delta = b^2 - 4ac$
• If $\Delta > 0$: Two real x-intercepts
• If $\Delta = 0$: One x-intercept (vertex on x-axis)
• If $\Delta < 0$: No real x-intercepts
Example 3: Find all intercepts of $f(x) = x^2 - 4x + 3$

Y-intercept: $f(0) = 3$ → $(0, 3)$

X-intercepts: Solve $x^2 - 4x + 3 = 0$
Factor: $(x - 1)(x - 3) = 0$
$x = 1$ or $x = 3$
X-intercepts: $(1, 0)$ and $(3, 0)$

3. Complete a Function Table: Quadratic Functions

Function Table: Shows input (x) and output (y) values
Example 1: Complete table for $f(x) = x^2 - 2x + 1$

x-10123
f(x)?????

Calculate:
$f(-1) = (-1)^2 - 2(-1) + 1 = 1 + 2 + 1 = 4$
$f(0) = 0 - 0 + 1 = 1$
$f(1) = 1 - 2 + 1 = 0$
$f(2) = 4 - 4 + 1 = 1$
$f(3) = 9 - 6 + 1 = 4$

x-10123
f(x)41014

Notice symmetry around $x = 1$ (axis of symmetry)!

4. Transformations of Quadratic Functions

Transformation: Changes to the parent function $f(x) = x^2$
General Form: $g(x) = a(x - h)^2 + k$
Four Types of Transformations:

1. Vertical Stretch/Compression (parameter $a$):
• If $|a| > 1$: Vertical stretch (narrower)
• If $0 < |a| < 1$: Vertical compression (wider)
• If $a < 0$: Reflection over x-axis

2. Horizontal Shift (parameter $h$):
• $f(x - h)$: Shift RIGHT by $h$ units
• $f(x + h)$: Shift LEFT by $h$ units
Opposite of sign!

3. Vertical Shift (parameter $k$):
• $f(x) + k$: Shift UP by $k$ units
• $f(x) - k$: Shift DOWN by $k$ units

4. Reflection:
• $-f(x)$: Reflect over x-axis (flip upside down)
Example 1: Describe transformations of $g(x) = 2(x - 3)^2 + 5$ from parent $f(x) = x^2$

$a = 2$: Vertical stretch by factor of 2 (narrower)
$h = 3$: Horizontal shift RIGHT 3 units
$k = 5$: Vertical shift UP 5 units

Vertex moves from $(0, 0)$ to $(3, 5)$
Example 2: Describe transformations of $h(x) = -\frac{1}{2}(x + 4)^2 - 1$

$a = -\frac{1}{2}$:
• Reflection over x-axis (opens down)
• Vertical compression by $\frac{1}{2}$ (wider)

$h = -4$: Shift LEFT 4 units
$k = -1$: Shift DOWN 1 unit

Vertex: $(-4, -1)$
Remember: In $f(x - h)$, the transformation is OPPOSITE the sign!
• $(x - 5)$ means shift RIGHT 5
• $(x + 5)$ means shift LEFT 5

5. Graph Quadratic Functions in Vertex Form

Vertex Form:
$$f(x) = a(x - h)^2 + k$$

where:
• $(h, k)$ = vertex
• $x = h$ = axis of symmetry
• $a$ determines opening direction and width
Steps to Graph from Vertex Form:
Step 1: Identify vertex $(h, k)$
Step 2: Plot vertex
Step 3: Draw axis of symmetry $x = h$
Step 4: Find additional points (use symmetry)
Step 5: Draw smooth parabola
Step 6: Label key features
Example 1: Graph $f(x) = (x - 2)^2 - 3$

Step 1: Vertex = $(2, -3)$
Step 2: Axis of symmetry: $x = 2$
Step 3: Opens UP ($a = 1 > 0$)

Additional points:
$f(3) = (3-2)^2 - 3 = 1 - 3 = -2$ → $(3, -2)$
By symmetry: $(1, -2)$
$f(4) = 4 - 3 = 1$ → $(4, 1)$
By symmetry: $(0, 1)$

Key points: $(2, -3)$, $(3, -2)$, $(1, -2)$, $(4, 1)$, $(0, 1)$

6. Write a Quadratic Function in Vertex Form

Converting Standard to Vertex Form: Use completing the square
Steps to Complete the Square:
Step 1: Factor out $a$ from $x^2$ and $x$ terms
Step 2: Take half of coefficient of $x$, square it
Step 3: Add and subtract this value inside parentheses
Step 4: Factor perfect square trinomial
Step 5: Simplify to vertex form
Example 1: Convert $f(x) = x^2 + 6x + 5$ to vertex form

Step 1: Group x-terms
$f(x) = (x^2 + 6x) + 5$

Step 2: Complete the square
Half of 6 is 3, and $3^2 = 9$
$f(x) = (x^2 + 6x + 9 - 9) + 5$
$f(x) = (x^2 + 6x + 9) - 9 + 5$

Step 3: Factor and simplify
$f(x) = (x + 3)^2 - 4$

Answer: $f(x) = (x + 3)^2 - 4$
Vertex: $(-3, -4)$
Example 2: Convert $f(x) = 2x^2 - 12x + 7$ to vertex form

Step 1: Factor out $a = 2$
$f(x) = 2(x^2 - 6x) + 7$

Step 2: Complete the square
Half of -6 is -3, and $(-3)^2 = 9$
$f(x) = 2(x^2 - 6x + 9 - 9) + 7$
$f(x) = 2(x^2 - 6x + 9) - 2(9) + 7$
$f(x) = 2(x - 3)^2 - 18 + 7$
$f(x) = 2(x - 3)^2 - 11$

Answer: $f(x) = 2(x - 3)^2 - 11$
Vertex: $(3, -11)$

7. Graph Quadratic Functions in Standard Form

Steps to Graph $f(x) = ax^2 + bx + c$:
Step 1: Find vertex using $h = -\frac{b}{2a}$, then $k = f(h)$
Step 2: Find y-intercept: $(0, c)$
Step 3: Find x-intercepts (if they exist)
Step 4: Draw axis of symmetry: $x = h$
Step 5: Plot points and use symmetry
Step 6: Draw smooth parabola
Example 1: Graph $f(x) = x^2 - 4x + 3$

Vertex:
$h = -\frac{-4}{2(1)} = 2$
$k = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$
Vertex: $(2, -1)$

Y-intercept: $(0, 3)$

X-intercepts:
$x^2 - 4x + 3 = 0$
$(x - 1)(x - 3) = 0$
$x = 1$ and $x = 3$
Points: $(1, 0)$ and $(3, 0)$

Opens UP ($a = 1 > 0$)

8. Match Quadratic Functions and Graphs

Matching Strategy:

Check these features:
1. Direction: Opens up ($a > 0$) or down ($a < 0$)?
2. Vertex location: Where is the highest/lowest point?
3. Y-intercept: Value of $c$ in standard form
4. Width: Narrow ($|a| > 1$) or wide ($|a| < 1$)?
5. Axis of symmetry: Vertical line through vertex
Example: Match function to key features

Function: $f(x) = -2(x - 1)^2 + 4$

Features to look for in graph:
• Opens DOWN ($a = -2 < 0$)
• Vertex at $(1, 4)$
• Axis of symmetry: $x = 1$
• Narrower than $y = x^2$ ($|a| = 2 > 1$)
• Maximum value is 4

9-10. Domain and Range of Quadratic Functions

Domain: All possible x-values (input)
Range: All possible y-values (output)
For Quadratic Functions:

Domain: Always all real numbers
$$\text{Domain: } (-\infty, \infty) \text{ or } \{x | x \in \mathbb{R}\}$$

Range: Depends on vertex and direction

If opens UP ($a > 0$):
Minimum value at vertex $(h, k)$
$$\text{Range: } [k, \infty) \text{ or } \{y | y \geq k\}$$

If opens DOWN ($a < 0$):
Maximum value at vertex $(h, k)$
$$\text{Range: } (-\infty, k] \text{ or } \{y | y \leq k\}$$
Example 1: Find domain and range of $f(x) = (x - 3)^2 + 2$

Domain: All real numbers → $(-\infty, \infty)$

Vertex: $(3, 2)$
Opens UP: $a = 1 > 0$
Minimum value: $y = 2$

Range: $[2, \infty)$ or $y \geq 2$
Example 2: Find domain and range of $g(x) = -2x^2 + 8x - 3$

Domain: $(-\infty, \infty)$

Find vertex:
$h = -\frac{8}{2(-2)} = \frac{-8}{-4} = 2$
$k = -2(2)^2 + 8(2) - 3 = -8 + 16 - 3 = 5$
Vertex: $(2, 5)$

Opens DOWN: $a = -2 < 0$
Maximum value: $y = 5$

Range: $(-\infty, 5]$ or $y \leq 5$

11. Write Quadratic Function from Vertex and Another Point

Steps:
Step 1: Write vertex form: $f(x) = a(x - h)^2 + k$
Step 2: Substitute vertex $(h, k)$
Step 3: Substitute the other point to find $a$
Step 4: Write complete function
Example 1: Write function with vertex $(2, -3)$ and passes through $(4, 5)$

Step 1: Vertex form with $(h, k) = (2, -3)$
$f(x) = a(x - 2)^2 - 3$

Step 2: Use point $(4, 5)$
$5 = a(4 - 2)^2 - 3$
$5 = a(2)^2 - 3$
$5 = 4a - 3$
$8 = 4a$
$a = 2$

Answer: $f(x) = 2(x - 2)^2 - 3$
Example 2: Vertex at $(-1, 4)$, passes through $(1, 0)$

$f(x) = a(x + 1)^2 + 4$
$0 = a(1 + 1)^2 + 4$
$0 = 4a + 4$
$a = -1$

Answer: $f(x) = -(x + 1)^2 + 4$

12-13. Write Quadratic Function from X-intercepts and Another Point

Factored/Intercept Form:
$$f(x) = a(x - p)(x - q)$$
where $p$ and $q$ are x-intercepts (zeros)
Steps:
Step 1: Write $f(x) = a(x - p)(x - q)$ using x-intercepts
Step 2: Substitute the given point to find $a$
Step 3: Write complete function
Step 4: Expand to standard form if needed
Example 1: X-intercepts at $x = 1$ and $x = 5$, passes through $(3, -4)$

Step 1: Factored form
$f(x) = a(x - 1)(x - 5)$

Step 2: Use point $(3, -4)$
$-4 = a(3 - 1)(3 - 5)$
$-4 = a(2)(-2)$
$-4 = -4a$
$a = 1$

Factored form: $f(x) = (x - 1)(x - 5)$

Standard form:
$f(x) = x^2 - 5x - x + 5$
$f(x) = x^2 - 6x + 5$
Example 2: Zeros at $x = -2$ and $x = 3$, passes through $(0, 6)$

$f(x) = a(x + 2)(x - 3)$
$6 = a(0 + 2)(0 - 3)$
$6 = a(2)(-3)$
$6 = -6a$
$a = -1$

Factored: $f(x) = -(x + 2)(x - 3)$
Standard: $f(x) = -x^2 + x + 6$

14. Interpret Parts of Quadratic Expressions: Word Problems

Real-World Applications:
• Projectile motion (height vs time)
• Area problems
• Profit/revenue models
• Optimization problems

Projectile Motion

Height Formula:
$$h(t) = -16t^2 + v_0t + h_0$$

where:
• $h(t)$ = height at time $t$ (feet)
• $t$ = time (seconds)
• $v_0$ = initial velocity (ft/s)
• $h_0$ = initial height (feet)
• $-16$ comes from gravity ($-\frac{1}{2}g$ where $g = 32$ ft/s²)
Example 1: A ball is thrown upward from a height of 5 feet with initial velocity of 48 ft/s.
Equation: $h(t) = -16t^2 + 48t + 5$

Interpret parts:
• $-16$: gravity's effect (always negative)
• $48$: initial upward velocity in ft/s
• $5$: starting height in feet

Find maximum height:
$t = -\frac{48}{2(-16)} = \frac{48}{32} = 1.5$ seconds
$h(1.5) = -16(1.5)^2 + 48(1.5) + 5$
$h(1.5) = -36 + 72 + 5 = 41$ feet

Maximum height: 41 feet at 1.5 seconds

Revenue/Profit Models

Example 2: A company's profit (in thousands) is modeled by:
$P(x) = -2x^2 + 80x - 200$
where $x$ is the number of items sold (in hundreds)

Interpret:
• $-2$: Profit decreases as production increases beyond optimal point
• $80$: Rate of profit increase per 100 items
• $-200$: Fixed costs (startup costs)

Maximum profit:
$x = -\frac{80}{2(-2)} = 20$ (hundreds of items = 2000 items)
$P(20) = -2(20)^2 + 80(20) - 200$
$P(20) = -800 + 1600 - 200 = 600$ thousand = $600,000

Maximum profit: $600,000 when 2000 items sold

Area Problems

Example 3: A rectangular garden has length $(x + 5)$ and width $(x + 3)$.
Find area function and dimensions for maximum area if perimeter is 32 feet.

Area:
$A(x) = (x + 5)(x + 3)$
$A(x) = x^2 + 8x + 15$

Perimeter constraint:
$2(x + 5) + 2(x + 3) = 32$
$2x + 10 + 2x + 6 = 32$
$4x + 16 = 32$
$x = 4$

Dimensions: Length = 9 ft, Width = 7 ft
Area: $A = 9 \times 7 = 63$ square feet

Summary: Forms of Quadratic Functions

FormEquationWhat It ShowsHow to Find
Standard Form$f(x) = ax^2 + bx + c$Y-intercept $(0, c)$Expand from other forms
Vertex Form$f(x) = a(x - h)^2 + k$Vertex $(h, k)$
Axis: $x = h$
Complete the square
Factored Form$f(x) = a(x - p)(x - q)$X-intercepts $p$ and $q$Factor or use zeros

Key Formulas Reference

FeatureFormulaNotes
Vertex (x-coordinate)$h = -\frac{b}{2a}$From standard form
Vertex (y-coordinate)$k = f(h)$Substitute $h$ into function
Axis of Symmetry$x = -\frac{b}{2a}$ or $x = h$Vertical line through vertex
Y-intercept$(0, c)$From standard form
X-intercepts$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$Quadratic formula
Discriminant$\Delta = b^2 - 4ac$Determines number of roots

Quick Checklist for Graphing

Before Graphing, Identify:

Direction: $a > 0$ (up) or $a < 0$ (down)?
Vertex: $(h, k)$ - highest or lowest point
Axis of Symmetry: $x = h$
Y-intercept: $(0, c)$ or $f(0)$
X-intercepts: Factor or use quadratic formula
Width: Narrow ($|a| > 1$) or wide ($|a| < 1$)?
Domain: Always $(-\infty, \infty)$
Range: $[k, \infty)$ if up, $(-\infty, k]$ if down
Success Tips for Quadratic Functions:
✓ Memorize vertex formula: $h = -\frac{b}{2a}$
✓ Know all three forms and when to use each
✓ Vertex form shows vertex directly: $(h, k)$
✓ Standard form shows y-intercept: $c$
✓ Factored form shows x-intercepts: $p$ and $q$
✓ Sign of $a$ determines if opens up or down
✓ Domain is ALWAYS all real numbers
✓ Range depends on vertex and direction
✓ Use completing the square to convert to vertex form
✓ Check your work by substituting points back into equation
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