7.3 Quadratic Graphs, Parabolas & Basic Non-Linear Functions
Key Takeaways
- The graph of a quadratic function $f(x) = ax^2 + bx + c$ is a symmetric U-shaped curve called a parabola that opens upward when $a > 0$ (having a minimum) and downward when $a < 0$ (having a maximum).
- The vertical axis of symmetry is given by the formula $x = -\frac{b}{2a}$, and the vertex coordinates are $(h, k) = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$.
- In vertex form $y = a(x - h)^2 + k$, the vertex $(h, k)$ is immediately visible, where $k$ is the absolute maximum value (if $a < 0$) or minimum value (if $a > 0$) of the function.
- In projectile motion models $h(t) = -16t^2 + v_0 t + h_0$, the vertex $(t_v, h_v)$ represents the peak height and time to peak, while solving $h(t) = 0$ yields the total flight time until ground impact.
- Linear models grow by constant additive increments (constant first differences), quadratic models have constant second differences, and exponential models ($y = a \cdot b^x$) grow by constant multiplicative factors (constant ratios).
Quadratic Graphs, Parabolas & Basic Non-Linear Functions
Quick Summary: A quadratic function produces a smooth, symmetric U-shaped curve called a parabola. The turning point of the curve is the vertex, representing the function's global minimum (if the parabola opens upward) or global maximum (if it opens downward). Non-linear models differ fundamentally from linear functions: while linear equations change by a constant additive slope, quadratics change with variable rate (constant second differences), and exponential functions change by a constant multiplicative percentage ratio.
Graph interpretation on the HiSET focuses on locating vertices, finding lines of symmetry, identifying intercepts, modeling projectile flight, and differentiating between linear, quadratic, and exponential relationships.
Anatomy of a Parabola
A quadratic function can be expressed in three primary algebraic forms:
- Standard Form: $y = ax^2 + bx + c$
- Vertex Form: $y = a(x - h)^2 + k$ (where $(h, k)$ is the vertex)
- Factored / Intercept Form: $y = a(x - r_1)(x - r_2)$ (where $r_1, r_2$ are the $x$-intercepts)
Key Anatomical Features of a Parabola
Opens Upward (a > 0) Opens Downward (a < 0)
y y
│ │ Axis of Symmetry │ ▲ Vertex (h, k) [MAXIMUM]
│ \ │ / │ /│\
│ \ │ / │ / │ \
y-int ───►• \ │ / │ / │ \ ◄── Parabola Curve
(0, c) │ \ │ / │ / │ \
─────┼──────•─┼─•────── x ─────┼───•────┼────•─── x
│ r₁│ r₂ │ r₁ │ r₂
│ ▼ │ │
│ Vertex (h, k) [MINIMUM] │ │ Axis of Symmetry: x = -b/(2a)
Core Properties of Parabolas
- Direction of Opening:
- If $a > 0$, the parabola opens upward ($\bigcup$), and the vertex is the minimum point.
- If $a < 0$, the parabola opens downward ($\bigcap$), and the vertex is the maximum point.
- Axis of Symmetry: The vertical line passing directly through the vertex, dividing the parabola into symmetric mirror halves:
- Vertex Coordinates $(h, k)$:
- $y$-Intercept: Found by evaluating $f(0)$: the point is always $(0, c)$.
- $x$-Intercepts (Roots / Zeros): Found by solving $ax^2 + bx + c = 0$ using factoring or the Quadratic Formula.
Worked Example: Graph Analysis from Standard Form
Analyze the quadratic function: $f(x) = -x^2 + 6x - 5$
- Direction of Opening: Leading coefficient $a = -1 < 0$. The parabola opens downward and has a maximum value.
- Axis of Symmetry: The axis of symmetry is the vertical line $x = 3$.
- Vertex $(h, k)$: Substitute $x = 3$ into $f(x)$: The vertex is $(3, 4)$, and the maximum value of the function is $4$.
- $y$-Intercept:
- $x$-Intercepts (Zeros): Set $f(x) = 0$: The $x$-intercepts are $(1, 0)$ and $(5, 0)$.
Real-World Applications: Projectile Motion & Optimization
On the HiSET, quadratic word problems frequently involve physics (objects thrown, dropped, or launched) and business economics (revenue maximization).
1. Projectile Motion Formula
When an object is launched vertically near Earth's surface, its height $h(t)$ in feet after $t$ seconds is modeled by:
- $-16t^2$: Gravitational acceleration term (in feet per second squared).
- $v_0$: Initial vertical velocity (in $\text{ft/s}$).
- $h_0$: Initial launch height (in feet at $t = 0$).
Projectile Motion Flight Path
Height h(t)
▲ Vertex: Peak Height (t_max, h_max)
│ •
│ / \
│ / \
Initial ──►• h₀ / \
Height │ / \
│ / \
────────────┼─────────•───────────────────────•─────► Time (t)
0 Launch Ground Hit (h = 0)
Worked Example: Projectile Flight
A toy rocket is launched from ground level ($h_0 = 0$) with an initial upward velocity of $64\text{ ft/s}$. Its height function is $h(t) = -16t^2 + 64t$.
- At what time does the rocket reach its maximum height?
- What is the maximum height reached?
- When does the rocket hit the ground? Set $h(t) = 0$:
Comparing Non-Linear Models: Linear vs. Quadratic vs. Exponential
The HiSET requires distinguishing between mathematical models based on formulas, graphs, and data tables:
Model Comparison Matrix
| Function Type | Algebraic Equation | Graph Shape | Rate of Change Pattern | Real-World Context |
|---|---|---|---|---|
| Linear | $y = mx + b$ | Straight line | Constant additive difference (First differences are equal) | Constant speed, hourly wages, simple interest |
| Quadratic | $y = ax^2 + bx + c$ | U-shaped parabola | Non-constant slope (Second differences are equal) | Projectile trajectories, area optimization, gravity |
| Exponential | $y = a \cdot b^x$ | Curved J-shape | Constant multiplicative ratio (Multiplies by factor $b$) | Population growth, bacterial division, compound interest, radioactive decay |
Identifying Models from Numerical Tables
To identify a function type from sequential $x$-values ($x = 0, 1, 2, 3, 4$):
Table Analysis: Differences vs. Ratios
x │ Linear y First Diff x │ Quadratic y 1st Diff 2nd Diff x │ Exponential y Ratio (y₂/y₁)
──┼───────── ────────── ──┼──────────── ──────── ──────── ──┼────────────── ─────────────
0 │ 4 0 │ 1 0 │ 3
1 │ 7 +3 1 │ 4 +3 1 │ 6 ×2
2 │ 10 +3 2 │ 9 +5 +2 2 │ 12 ×2
3 │ 13 +3 3 │ 16 +7 +2 3 │ 24 ×2
4 │ 16 +3 4 │ 25 +9 +2 4 │ 48 ×2
Exponential Growth and Decay Formulas
- $a$: Initial amount (at $t = 0$).
- $r$: Growth or decay rate as a decimal (e.g., $5% = 0.05$).
- $b = 1 \pm r$: Growth factor ($1 + r$) or decay factor ($1 - r$).
What is the vertex and the axis of symmetry for the quadratic function f(x) = -x^2 + 6x - 5?
A projectile is launched upward with its height modeled by h(t) = -16t^2 + 64t, where h is height in feet and t is time in seconds. What is the maximum height reached by the projectile?
A dataset contains the values (0, 3), (1, 6), (2, 12), (3, 24), and (4, 48). Which mathematical model best describes this relationship?
The height of an architectural arch is modeled by the equation y = -0.5(x - 10)^2 + 50, where x is the horizontal distance from the left edge in feet and y is height in feet. What is the maximum height of the arch and its total span (width) along the ground?