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: $f(x) = x^2$
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. 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
• 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)
• 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: $(h, k)$
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: $(2, -5)$
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
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: $x = 3$
$a = -1$, $b = 6$
$x = -\frac{6}{2(-1)} = -\frac{6}{-2} = 3$
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
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 $(3, 0)$
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 $f(x) = x^2 - 2x + 1$
Calculate:
$f(-1) = (-1)^2 - 2(-1) + 1 = 1 + 2 + 1 = 4$
$f(0) = 0 - 0 + 1 = 1$
$f(1) = 1 - 2 + 1 = 0$
$f(2) = 4 - 4 + 1 = 1$
$f(3) = 9 - 6 + 1 = 4$
Notice symmetry around $x = 1$ (axis of symmetry)!
x | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|
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 | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|
f(x) | 4 | 1 | 0 | 1 | 4 |
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: $g(x) = a(x - h)^2 + k$
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)
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 $(3, 5)$
$a = 2$: 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: $(-4, -1)$
$a = -\frac{1}{2}$:
• 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
• $(x - 5)$ 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
$$f(x) = a(x - h)^2 + k$$
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
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: $(2, -3)$, $(3, -2)$, $(1, -2)$, $(4, 1)$, $(0, 1)$
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
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: $(-3, -4)$
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: $(3, -11)$
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
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 ($a = 1 > 0$)
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
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
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)
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 $(h, k)$
$$\text{Range: } (-\infty, k] \text{ or } \{y | y \leq k\}$$
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 $y \geq 2$
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 $y \leq 5$
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
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: $f(x) = 2(x - 2)^2 - 3$
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: $f(x) = -(x + 1)^2 + 4$
$f(x) = a(x + 1)^2 + 4$
$0 = a(1 + 1)^2 + 4$
$0 = 4a + 4$
$a = -1$
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)
$$f(x) = a(x - p)(x - q)$$
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
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:
$f(x) = x^2 - 5x - x + 5$
$f(x) = x^2 - 6x + 5$
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: $f(x) = -x^2 + x + 6$
$f(x) = a(x + 2)(x - 3)$
$6 = a(0 + 2)(0 - 3)$
$6 = a(2)(-3)$
$6 = -6a$
$a = -1$
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 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)
$$h(t) = -16t^2 + v_0t + h_0$$
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
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
$P(x) = -2x^2 + 80x - 200$
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
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
Form | Equation | What It Shows | How 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
Feature | Formula | Notes |
---|---|---|
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
☐ 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
✓ 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