5.2 Non-Linear Graphs
What are Non-Linear Graphs?
Non-linear graphs are curves, not straight lines. They represent equations where the highest power of $x$ is greater than 1, or where $x$ appears in other forms (like in the denominator or as an exponent).
Non-linear graphs are curves, not straight lines. They represent equations where the highest power of $x$ is greater than 1, or where $x$ appears in other forms (like in the denominator or as an exponent).
Quadratic
⌣
$y = x^2$
Parabola (U-shape)Cubic
∿
$y = x^3$
S-shape curveReciprocal
⌒ ⌒
$y = \frac{1}{x}$
HyperbolaExponential
$y = a^x$
Rapid growth/decay
🎯 Which graph type?
The Parabola
Quadratic graphs create a parabola - a smooth U-shaped curve (or ∩-shaped if $a$ is negative).
Quadratic graphs create a parabola - a smooth U-shaped curve (or ∩-shaped if $a$ is negative).
⚡ Key Features of a Parabola:
• Vertex (turning point): The minimum or maximum point
• Line of symmetry: Vertical line through the vertex
• y-intercept: Where the curve crosses the y-axis (at $x=0$, $y=c$)
• Roots/x-intercepts: Where the curve crosses the x-axis
• Line of symmetry: Vertical line through the vertex
• y-intercept: Where the curve crosses the y-axis (at $x=0$, $y=c$)
• Roots/x-intercepts: Where the curve crosses the x-axis
💡 Effect of the Coefficient $a$:
• If $a > 0$: Parabola opens upward ⌣ (minimum point)
• If $a < 0$: Parabola opens downward ⌢ (maximum point)
• Larger $|a|$: Narrower parabola
• Smaller $|a|$: Wider parabola
• If $a < 0$: Parabola opens downward ⌢ (maximum point)
• Larger $|a|$: Narrower parabola
• Smaller $|a|$: Wider parabola
Quadratic Graph Constructor
y = x²
Example: Sketch $y = x^2 - 4x + 3$
Step 1: Find the y-intercept (set $x = 0$)
$y = 0 - 0 + 3 = 3$ → Point: $(0, 3)$
Step 2: Find the roots (set $y = 0$)
$x^2 - 4x + 3 = 0$
$(x - 1)(x - 3) = 0$
$x = 1$ or $x = 3$ → Points: $(1, 0)$ and $(3, 0)$
Step 3: Find the vertex
Line of symmetry: $x = \frac{1 + 3}{2} = 2$
$y = 4 - 8 + 3 = -1$ → Vertex: $(2, -1)$
Step 4: Since $a = 1 > 0$, parabola opens upward
$y = 0 - 0 + 3 = 3$ → Point: $(0, 3)$
Step 2: Find the roots (set $y = 0$)
$x^2 - 4x + 3 = 0$
$(x - 1)(x - 3) = 0$
$x = 1$ or $x = 3$ → Points: $(1, 0)$ and $(3, 0)$
Step 3: Find the vertex
Line of symmetry: $x = \frac{1 + 3}{2} = 2$
$y = 4 - 8 + 3 = -1$ → Vertex: $(2, -1)$
Step 4: Since $a = 1 > 0$, parabola opens upward
🎯 Find the y-intercept
🎯 Up or Down?
The Cubic Curve
Cubic graphs have an S-shape (or reverse S-shape). They can have up to 3 roots and 2 turning points.
Cubic graphs have an S-shape (or reverse S-shape). They can have up to 3 roots and 2 turning points.
⚡ Key Features of Cubic Graphs:
• If $a > 0$: Goes from bottom-left to top-right ↗
• If $a < 0$: Goes from top-left to bottom-right ↘
• Has a point of inflection (where curvature changes)
• Can cross the x-axis 1, 2, or 3 times
• If $a < 0$: Goes from top-left to bottom-right ↘
• Has a point of inflection (where curvature changes)
• Can cross the x-axis 1, 2, or 3 times
Cubic Graph Constructor
y = x³
Common Cubic Shapes:
$y = x^3$: Basic S-curve through origin
$y = x^3 - 3x$: Has two turning points (local max and min)
$y = (x-1)(x-2)(x-3)$: Crosses x-axis at 1, 2, and 3
$y = x^3 - 3x$: Has two turning points (local max and min)
$y = (x-1)(x-2)(x-3)$: Crosses x-axis at 1, 2, and 3
🎯 Which direction does it go?
The Hyperbola
Reciprocal graphs create a hyperbola - two separate curves that never touch the axes.
Reciprocal graphs create a hyperbola - two separate curves that never touch the axes.
⚡ Key Features of Reciprocal Graphs:
• Asymptotes: Lines the curve approaches but never touches
- Vertical asymptote at $x = 0$ (y-axis)
- Horizontal asymptote at $y = 0$ (x-axis)
• Never passes through the origin
• Has two separate branches
- Vertical asymptote at $x = 0$ (y-axis)
- Horizontal asymptote at $y = 0$ (x-axis)
• Never passes through the origin
• Has two separate branches
💡 Effect of $k$:
• If $k > 0$: Curves in quadrants I and III
• If $k < 0$: Curves in quadrants II and IV
• Larger $|k|$: Curves further from origin
• If $k < 0$: Curves in quadrants II and IV
• Larger $|k|$: Curves further from origin
Reciprocal Graph Constructor
y = 1/x
Example: Key points for $y = \frac{2}{x}$
When $x = 1$: $y = 2$ → Point: $(1, 2)$
When $x = 2$: $y = 1$ → Point: $(2, 1)$
When $x = -1$: $y = -2$ → Point: $(-1, -2)$
When $x = -2$: $y = -1$ → Point: $(-2, -1)$
Notice: The curve is symmetric about the origin
When $x = 2$: $y = 1$ → Point: $(2, 1)$
When $x = -1$: $y = -2$ → Point: $(-1, -2)$
When $x = -2$: $y = -1$ → Point: $(-2, -1)$
Notice: The curve is symmetric about the origin
🎯 Which quadrants?
Exponential Growth and Decay
Exponential graphs show rapid growth or decay. They're used to model population growth, radioactive decay, and compound interest.
Exponential graphs show rapid growth or decay. They're used to model population growth, radioactive decay, and compound interest.
⚡ Key Features of Exponential Graphs:
• Always passes through $(0, 1)$ (since $a^0 = 1$)
• Horizontal asymptote at $y = 0$
• Never touches or crosses the x-axis
• Always positive (above x-axis)
• Horizontal asymptote at $y = 0$
• Never touches or crosses the x-axis
• Always positive (above x-axis)
💡 Effect of the Base $a$:
• If $a > 1$: Exponential growth (curve rises steeply to the right)
• If $0 < a < 1$: Exponential decay (curve falls towards zero)
• Larger base = steeper growth
• $y = a^{-x}$ is the same as $y = (\frac{1}{a})^x$
• If $0 < a < 1$: Exponential decay (curve falls towards zero)
• Larger base = steeper growth
• $y = a^{-x}$ is the same as $y = (\frac{1}{a})^x$
Exponential Graph Constructor
y = 2ˣ
Example: Key points for $y = 2^x$
When $x = -2$: $y = 2^{-2} = \frac{1}{4} = 0.25$
When $x = -1$: $y = 2^{-1} = \frac{1}{2} = 0.5$
When $x = 0$: $y = 2^0 = 1$
When $x = 1$: $y = 2^1 = 2$
When $x = 2$: $y = 2^2 = 4$
When $x = 3$: $y = 2^3 = 8$
Notice how quickly it grows!
When $x = -1$: $y = 2^{-1} = \frac{1}{2} = 0.5$
When $x = 0$: $y = 2^0 = 1$
When $x = 1$: $y = 2^1 = 2$
When $x = 2$: $y = 2^2 = 4$
When $x = 3$: $y = 2^3 = 8$
Notice how quickly it grows!
🎯 Growth or Decay?
Real Life Uses:
• Finance: Compound interest
• Physics: Radioactive decay (half-life)
• Medicine: Drug concentration in bloodstream
• Finance: Compound interest
• Physics: Radioactive decay (half-life)
• Medicine: Drug concentration in bloodstream
Transforming Graphs
You can transform any graph by adding, subtracting, or multiplying. Understanding these transformations helps you sketch graphs quickly.
You can transform any graph by adding, subtracting, or multiplying. Understanding these transformations helps you sketch graphs quickly.
⚡ Transformation Rules:
Translations (shifts):
• $y = f(x) + a$: Move up by $a$ units
• $y = f(x) - a$: Move down by $a$ units
• $y = f(x + a)$: Move left by $a$ units
• $y = f(x - a)$: Move right by $a$ units
Inside the bracket = opposite direction!
• $y = f(x) + a$: Move up by $a$ units
• $y = f(x) - a$: Move down by $a$ units
• $y = f(x + a)$: Move left by $a$ units
• $y = f(x - a)$: Move right by $a$ units
Inside the bracket = opposite direction!
💡 Stretches and Reflections:
Stretches:
• $y = af(x)$: Vertical stretch by factor $a$
• $y = f(ax)$: Horizontal stretch by factor $\frac{1}{a}$
Reflections:
• $y = -f(x)$: Reflect in the x-axis
• $y = f(-x)$: Reflect in the y-axis
• $y = af(x)$: Vertical stretch by factor $a$
• $y = f(ax)$: Horizontal stretch by factor $\frac{1}{a}$
Reflections:
• $y = -f(x)$: Reflect in the x-axis
• $y = f(-x)$: Reflect in the y-axis
Transformations
Starting with $y = x^2$, apply transformations:
y = x²
Example 1: Describe the transformation from $y = x^2$ to $y = (x - 3)^2 + 2$
• $(x - 3)$ means translate 3 units right
• $+ 2$ means translate 2 units up
Answer: Translation by vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
• $+ 2$ means translate 2 units up
Answer: Translation by vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$
Example 2: What is the equation of $y = x^2$ after reflecting in the x-axis and moving up 4?
• Reflect in x-axis: $y = -x^2$
• Move up 4: $y = -x^2 + 4$
Answer: $y = -x^2 + 4$ (or $y = 4 - x^2$)
• Move up 4: $y = -x^2 + 4$
Answer: $y = -x^2 + 4$ (or $y = 4 - x^2$)
🎯 Describe the Transformation
🎯 What's the new equation?
Summary: How to Identify Graph Types
⚡ Quick Recognition Guide:
Linear ($y = mx + c$): Straight line
Quadratic ($y = ax^2 + ...$): U or ∩ shaped parabola
Cubic ($y = ax^3 + ...$): S-shaped curve
Reciprocal ($y = \frac{k}{x}$): Two separate curves (hyperbola)
Exponential ($y = a^x$): One-sided rapid growth/decay, asymptote at y=0
Quadratic ($y = ax^2 + ...$): U or ∩ shaped parabola
Cubic ($y = ax^3 + ...$): S-shaped curve
Reciprocal ($y = \frac{k}{x}$): Two separate curves (hyperbola)
Exponential ($y = a^x$): One-sided rapid growth/decay, asymptote at y=0
📊 Graph Gallery
Click a button to see the graph
🎯 Match the Equation to the Graph Type