Quadratic Characteristics Slides
FL B.E.S.T. Standard • MA.912.AR.3.6 & MA.912.AR.3.7
ALGEBRA 2
Characteristics of
Quadratic Functions
Connecting Intercepts, Zeros, Vertex, Axis of Symmetry, and Extrema Across Algebraic and Graphical Models.
Key Domain 1: Intercepts & Zeros
Connecting \(x\)-intercepts, function zeros & roots
Key Domain 2: Vertex & Symmetry
Axis of symmetry, extrema & range boundaries
Conceptual Overview
Anatomy of a Parabola
\( f(x) = ax^2 + bx + c \)
Case 1: \( a > 0 \) Opens Upward
When the leading coefficient is positive, the arms extend toward \( +\infty \).
Vertex is a Minimum: lowest \(y\)-value
Range: \( [k, \infty) \) where \( k = f(h) \)
Shape: "U-Shaped" Valley (\(\cup\))
Case 2: \( a < 0 \) Opens Downward
When the leading coefficient is negative, the arms extend toward \( -\infty \).
Vertex is a Maximum: highest \(y\)-value
Range: \( (-\infty, k] \) where \( k = f(h) \)
Shape: "Hill" Peak (\(\cap\))
In both cases, parabolas exhibit bilateral reflectional symmetry across the vertical axis of symmetry.
Structural Landmark
Axis of Symmetry & The Vertex
Vertical Mirror Line
Axis of Symmetry
The vertical line that bisects the parabola into two matching mirror halves.
Standard Form Formula: \( x = -\frac{b}{2a} \)
Symmetry Rule: Any two points at equal height \(y\) are equidistant from this line.
The Vertex: \( (h, k) \)
The single turning point of the curve lying directly on the axis of symmetry.
Vertex Coordinates: \( (h, k) = \left(-\frac{b}{2a}, \; f(h)\right) \)
Vertex Form: In \( f(x) = a(x - h)^2 + k \), the vertex \( (h, k) \) is identified immediately!
Exam Warning: The axis of symmetry is always an equation of a vertical line (\( x = h \)), never just a naked number!
Function Behavior
Maximum vs. Minimum Values
Optimization & Range
Value vs. Location
Florida B.E.S.T. standards strictly distinguish between the extremum value and where it occurs:
Extremum Value (The "What"): The \(y\)-coordinate: \( y = k \)
Extremum Location (The "Where"): The \(x\)-coordinate: \( x = h \)
Domain & Range Rules
Domain of ALL Quadratics:
\( (-\infty, \infty) \) or all real numbers
Range if \( a > 0 \) (Minimum):
\( [k, \infty) \quad \text{or} \quad y \ge k \)
Range if \( a < 0 \) (Maximum):
\( (-\infty, k] \quad \text{or} \quad y \le k \)
Real-World Model: Projectile maximum height is the vertex \(y\)-value; time to peak is the vertex \(x\)-value.
Boundary Landmark
The \(y\)-Intercept & Its Mirror Point
Input \( x = 0 \)
Algebraic Definition
The point where the parabola crosses the vertical \(y\)-axis. Set \( x = 0 \) in any quadratic form:
Standard Form: \( f(x) = ax^2 + bx + c \) \[ f(0) = a(0)^2 + b(0) + c = c \]
Therefore, the \(y\)-intercept is always the coordinate pair: (0, c).
Symmetric Partner Point
Because parabolas are symmetric across \( x = h \), the \(y\)-intercept always has a mirror point:
Distance to Axis of Symmetry: \( |h - 0| = h \) units away
Symmetric Mirror Coordinate: \( (2h, c) \) at the same height \( y = c \)
Fundamental Fact: Every quadratic function has exactly one \(y\)-intercept because its domain includes \( x = 0 \).
B.E.S.T. Core Concept Bridge
The Quad-Equivalence: 4 Ways to Say the Same Thing
\( f(x) = 0 \)
GRAPHICAL
\(x\)-Intercepts
Points where the graph intersects the horizontal \(x\)-axis.
\( (r_1, 0) \text{ and } (r_2, 0) \)
FUNCTION
Zeros
Input values \(x\) that yield an output value of zero.
\( f(x) = 0 \)
EQUATION
Roots
Solutions satisfying the quadratic equation.
\( ax^2 + bx + c = 0 \)
ALGEBRAIC
Factors
Linear binomial factors in factored form.
\( (x - r_1)(x - r_2) \)
The Discriminant \( b^2 - 4ac \): Predicts count of real \(x\)-intercepts:
\( \Delta > 0 \implies 2 \text{ intercepts} \) \( \Delta = 0 \implies 1 \text{ vertex touch} \) \( \Delta < 0 \implies 0 \text{ real intercepts} \)
Guided Example
Connecting All Characteristics in Action
\( f(x) = -x^2 + 4x + 5 \)
1
Orientation & AOS
• \( a = -1 < 0 \implies \) Opens DOWN
• Axis of Symmetry:
\[ x = -\frac{4}{2(-1)} = 2 \]
Vertical line: \( x = 2 \)
2
Vertex & Range
• Evaluate at \( x = 2 \):
\( f(2) = -(2)^2 + 4(2) + 5 = 9 \)
• Vertex: (2, 9)
• Maximum Value: \( 9 \)
• Range: \( (-\infty, 9] \)
3
All Intercepts
• \(y\)-Intercept: \( f(0) = 5 \implies \) (0, 5)
• Set \( f(x) = 0 \) to find zeros:
\( -(x - 5)(x + 1) = 0 \)
• Zeros: \( x = 5 \) and \( x = -1 \)
• \(x\)-Intercepts: \( (5, 0) \) and \( (-1, 0) \)
Symmetry Verification: Average of zeros = \( \frac{-1 + 5}{2} = 2 \), matching our Axis of Symmetry \( x = 2 \) perfectly!
Partner Challenge
Analyze: \( g(x) = 2x^2 - 8x - 10 \)
Active Retrieval
TARGET 1 • SYMMETRY & VERTEX
Find the Axis of Symmetry and the Vertex coordinates.
AOS: \( x = -\frac{-8}{2(2)} = 2 \) | Vertex: \( (2, -18) \)
TARGET 2 • EXTREMUM & RANGE
Does it have a max or min? State its value and the function range.
Minimum value = \(-18\) | Range: \( [-18, \infty) \)
TARGET 3 • Y-INTERCEPT
State the \(y\)-intercept and its symmetric mirror point.
\(y\)-int: \( (0, -10) \) | Mirror point: \( (4, -10) \)
TARGET 4 • ZEROS & EQUIVALENCE
Factor to find the zeros, roots, and \(x\)-intercepts.
\( 2(x - 5)(x + 1) = 0 \implies x = 5, -1 \implies (5, 0), (-1, 0) \)
Discussion Prompt: How does knowing the zeros \( x = 5 \) and \( x = -1 \) give you an instant alternative way to find the axis of symmetry?
B.E.S.T. Standards Synthesis
Quadratic Characteristics Master Summary
Quick Reference
Characteristic Standard Form: \( ax^2 + bx + c \) Key Conceptual Insight
Axis of Symmetry \( x = -\frac{b}{2a} \) Vertical line dividing the curve symmetrically
Vertex \( (h, k) \) \( (h, f(h)) \) Turning point; gives maximum or minimum
Extremum & Range \( a > 0 \implies \min k \quad | \quad a < 0 \implies \max k \) Value is \( k \); Range bounded by \( k \)
\(y\)-Intercept \( (0, c) \) Always exactly 1 \(y\)-intercept; mirror is \( (2h, c) \)
Zeros & \(x\)-Intercepts \( ax^2 + bx + c = 0 \) Zeros = Solutions = Roots = \(x\)-Intercept coords
Synthesis Reflection: Why is the axis of symmetry always exactly halfway between any two symmetric points on a quadratic curve?
