Quadratic Functions and Parabola Analysis: Study Guide

Quadratic Functions and Parabola Analysis

Forms of a Quadratic Equation

Quadratic functions are fundamental in algebra and are commonly written in two main forms. Understanding these forms is essential for analyzing and graphing parabolas.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex Form: $f(x) = a(x-h)^2 + k$

Key Properties:

• Orientation (Concavity):

  • If $a > 0$, the parabola opens upward (concave upward) and has a minimum point.

  • If $a < 0$, the parabola opens downward (concave downward) and has a maximum point.

• Vertex Coordinates:

  • From Standard Form: $x_v = -\frac{b}{2a}$, $y_v = f(x_v)$

  • From Vertex Form: The vertex is $(h, k)$, read directly from the equation.

• Axis of Symmetry: The vertical line passing through the vertex, given by $x = h$ or $x = -\frac{b}{2a}$.

• Domain: Always $(-\infty, +\infty)$ for any quadratic function.

• Range:

  • If $a > 0$: $[k, +\infty)$

  • If $a < 0$: $(-\infty, k]$

Standard 5-Step Method for Graphing Quadratic Functions

To analyze and graph any quadratic function, follow these five steps:

1. Determine Concavity/Orientation: Identify $a$ and state whether the parabola opens upward or downward.

2. Find the x-intercepts: Set $f(x) = 0$ and solve for $x$ (by factoring, taking square roots, or using the quadratic formula).

3. Find the y-intercept: Set $x = 0$ and evaluate $f(0)$.

4. Find the Vertex and Axis of Symmetry: Calculate $x_v = -\frac{b}{2a}$ (or read $h$), $y_v = f(x_v)$ (or read $k$), and state the axis of symmetry $x = h$.

5. Graph and State Domain & Range: Plot the vertex, intercepts, draw the symmetric parabola, and state the domain and range.

Examples: Step-by-Step Analysis

These examples illustrate the application of the 5-step method to various quadratic equations.

Example 1: Standard Form ($a > 0$)

• Equation: $f(x) = x^2 - 8x + 12$

• Step 1: Concavity $a = 1$ (positive) → Parabola opens upward.

• Step 2: x-intercepts $f(x) = 0 \rightarrow x^2 - 8x + 12 = 0$ Factor: $(x - 6)(x - 2) = 0 \rightarrow x = 6, x = 2$ x-intercepts: $(6, 0)$ and $(2, 0)$

• Step 3: y-intercept $x = 0 \rightarrow f(0) = 0^2 - 8 \cdot 0 + 12 = 12$ y-intercept: $(0, 12)$

• Step 4: Vertex & Axis of Symmetry $x_v = -\frac{-8}{2 \cdot 1} = 4$ $y_v = f(4) = 4^2 - 8 \cdot 4 + 12 = -4$ Vertex: $(4, -4)$ Axis of Symmetry: $x = 4$

• Step 5: Domain & Range Domain: $(-\infty, +\infty)$ Range: $[-4, +\infty)$

Example 2: Standard Form ($a < 0$, missing $b$ term)

• Equation: $f(x) = -x^2 + 4$

• Step 1: Concavity $a = -1$ (negative) → Parabola opens downward.

• Step 2: x-intercepts $f(x) = 0 \rightarrow -x^2 + 4 = 0 \rightarrow x^2 = 4 \rightarrow x = \pm 2$ x-intercepts: $(2, 0)$ and $(-2, 0)$

• Step 3: y-intercept $x = 0 \rightarrow f(0) = -0^2 + 4 = 4$ y-intercept: $(0, 4)$

• Step 4: Vertex & Axis of Symmetry $x_v = -\frac{0}{2 \cdot -1} = 0$ $y_v = f(0) = 4$ Vertex: $(0, 4)$ Axis of Symmetry: $x = 0$

• Step 5: Domain & Range Domain: $(-\infty, +\infty)$ Range: $(-\infty, 4]$

Example 3: Vertex Form ($a < 0$)

• Equation: $f(x) = -2(x-3)^2 + 8$

• Step 1: Concavity $a = -2$ (negative) → Parabola opens downward.

• Step 2: x-intercepts $f(x) = 0 \rightarrow -2(x-3)^2 + 8 = 0$ $-2(x-3)^2 = -8 \rightarrow (x-3)^2 = 4 \rightarrow x-3 = \pm 2 \rightarrow x = 3 \pm 2$ x-intercepts: $(5, 0)$ and $(1, 0)$

• Step 3: y-intercept $x = 0 \rightarrow f(0) = -2(0-3)^2 + 8 = -2 \cdot 9 + 8 = -18 + 8 = -10$ y-intercept: $(0, -10)$

• Step 4: Vertex & Axis of Symmetry From form: $h = 3$, $k = 8$ Vertex: $(3, 8)$ Axis of Symmetry: $x = 3$

• Step 5: Domain & Range Domain: $(-\infty, +\infty)$ Range: $(-\infty, 8]$

Example 4: Complex Intercepts

• Equation: $f(x) = (x+3)^2 + 1$

• Step 1: Concavity $a = 1$ (positive) → Parabola opens upward.

• Step 2: x-intercepts $f(x) = 0 \rightarrow (x+3)^2 + 1 = 0 \rightarrow (x+3)^2 = -1$ $(x+3) = \pm i$ No real x-intercepts (graph does not cross the x-axis).

• Step 3: y-intercept $x = 0 \rightarrow f(0) = (0+3)^2 + 1 = 9 + 1 = 10$ y-intercept: $(0, 10)$

• Step 4: Vertex & Axis of Symmetry From form: $h = -3$, $k = 1$ Vertex: $(-3, 1)$ Axis of Symmetry: $x = -3$

• Step 5: Domain & Range Domain: $(-\infty, +\infty)$ Range: $[1, +\infty)$

Real-World Application Problems

Quadratic functions are used to model various real-world scenarios, such as maximizing areas and projectile motion.

Example 5: Maximizing Fencing Area (3-Sided Enclosure)

• Problem: You have 600 ft of fencing to enclose a rectangular plot along a river (no fence along the river). Find dimensions that maximize area.

• Set up Equations:

  • Let width $= x$ (two width sides).

  • Length $= 600 - 2x$.

  • Area function: $A = (600 - 2x)x = -2x^2 + 600x$

• Find Maximum (Vertex): $x_v = -\frac{600}{2 \cdot -2} = 150$ ft

• Calculate Dimensions & Maximum Area:

  • Width: 150 ft

  • Length: $600 - 2 \cdot 150 = 300$ ft

  • Maximum Area: $150 \times 300 = 45,000$ sq ft

Example 6: Projectile Motion

• Problem: A ball is thrown upward/outward from a height of 6 ft. Its height is given by $f(x) = -0.1x^2 + 0.7x + 6$, where $x$ is horizontal distance in feet.

• Part A: Find Maximum Height and Distance Where It Occurs

  • Coefficients: $a = -0.1$, $b = 0.7$, $c = 6$

  • Horizontal distance of vertex: $x_{max} = -\frac{0.7}{2 \cdot -0.1} = 3.5$ ft

  • Maximum height: $f(3.5) = -0.1 \cdot (3.5)^2 + 0.7 \cdot 3.5 + 6 = 7.225$ ft

  • Answer: Maximum height is 7.225 ft, occurring 3.5 ft horizontally from where thrown.

• Part B: Total Horizontal Distance Until Ball Hits Ground

  • Set $f(x) = 0$: $-0.1x^2 + 0.7x + 6 = 0$

  • Multiply by $-10$: $x^2 - 7x - 60 = 0$

  • Factor: $(x - 12)(x + 5) = 0 \rightarrow x = 12$ or $x = -5$

  • Choose physical solution: $x = 12$ ft (distance cannot be negative)

  • Answer: The ball travels 12 ft horizontally before hitting the ground.

Summary Table: Quadratic Function Properties

Form	Vertex	Axis of Symmetry	Concavity	Domain	Range
Standard: $f(x) = ax^2 + bx + c$	$(x_v, y_v)$ $x_v = -\frac{b}{2a}$ $y_v = f(x_v)$	$x = -\frac{b}{2a}$	Upward if $a > 0$ Downward if $a < 0$	$(-\infty, +\infty)$	$[y_v, +\infty)$ if $a > 0$ $(-\infty, y_v]$ if $a < 0$
Vertex: $f(x) = a(x-h)^2 + k$	$(h, k)$	$x = h$	Upward if $a > 0$ Downward if $a < 0$	$(-\infty, +\infty)$	$[k, +\infty)$ if $a > 0$ $(-\infty, k]$ if $a < 0$

Additional info: All explanations have been expanded for clarity and completeness. The summary table is inferred for comparison purposes.