3.6 Graphs of Functions

Graphs of Functions

Graphs of functions are visual representations of how inputs relate to outputs. Being able to read and interpret these graphs is a core skill in algebra, since it lets you identify domain, range, intercepts, and behavior at a glance.

Graphs of Functions

Vertical Line Test for Functions

A function can only assign one output to each input. The vertical line test is a quick visual check that uses this rule: imagine sweeping a vertical line across the graph from left to right.

• If every vertical line you could draw hits the graph at most once, the graph represents a function.
• If any vertical line hits the graph more than once, the graph is not a function, because that x-value would have two different y-values.

For example, a circle fails the vertical line test. At most x-values within the circle, a vertical line crosses the curve in two places. A parabola that opens upward, on the other hand, passes the test since each vertical line touches it only once.

Graphs of Common Functions

Linear functions produce straight-line graphs.

• General form: $y = mx + b$
  • $m$ is the slope (steepness and direction). A positive $m$ means the line rises left to right; a negative $m$ means it falls.
  • $b$ is the y-intercept, the point $(0, b)$ where the line crosses the y-axis.

Quadratic functions produce parabolas, which are symmetric U-shaped curves.

• General form: $y = ax^2 + bx + c$, where $a \neq 0$
  • If $a > 0$, the parabola opens upward (the vertex is the lowest point).
  • If $a < 0$, the parabola opens downward (the vertex is the highest point).
• The turning point of the parabola is called the vertex. You can find its x-coordinate with $x = \frac{-b}{2a}$.

Exponential functions show rapid growth or decay with a characteristic curve.

• General form: $y = a \cdot b^x$, where $a \neq 0$ and $b > 0$, $b \neq 1$
  • If $b > 1$, the function models exponential growth (the graph rises steeply to the right).
  • If $0 < b < 1$, the function models exponential decay (the graph falls toward zero to the right).
• Exponential graphs have a horizontal asymptote, usually the x-axis, which the curve approaches but never touches.

Piecewise functions combine different rules for different intervals of the domain. Their graphs may look like segments of different function types stitched together. Pay attention to open vs. closed dots at the endpoints of each piece, since those tell you which piece "owns" that x-value.

Key Information from Function Graphs

Domain and Range

• The domain is the set of all x-values the graph covers. Read it by looking at how far left and right the graph extends.
• The range is the set of all y-values the graph covers. Read it by looking at how far up and down the graph extends.

For example, a parabola opening upward with vertex at $(2, -3)$ has a range of $y \geq -3$, because the graph never dips below that vertex.

Intercepts

• x-intercepts (also called zeros or roots) are where the graph crosses the x-axis, meaning $y = 0$.
• The y-intercept is where the graph crosses the y-axis, meaning $x = 0$. A function can have at most one y-intercept.

Increasing, Decreasing, and Constant Intervals

• The function is increasing on an interval where the graph goes up as you move left to right.
• The function is decreasing on an interval where the graph goes down as you move left to right.
• The function is constant on an interval where the graph stays flat.

These intervals are always described in terms of x-values. For instance, you'd say "the function is increasing on $(-\infty, 3)$," not "the function is increasing from $y = 1$ to $y = 5$."

Positive and Negative Regions

• The function is positive where the graph sits above the x-axis ($y > 0$).
• The function is negative where the graph sits below the x-axis ($y < 0$).

End Behavior

End behavior describes what happens to the y-values as x heads toward $+\infty$ or $-\infty$. For example, with $y = x^2$, as $x \to +\infty$ the graph rises without bound, and as $x \to -\infty$ it also rises without bound.

Function Transformations

Transformations let you take a parent function (like $y = x^2$) and shift, stretch, or flip it to create a new graph. Starting from $y = f(x)$:

• Vertical shift: $y = f(x) + k$ moves the graph up $k$ units (if $k > 0$) or down (if $k < 0$).
• Horizontal shift: $y = f(x - h)$ moves the graph right $h$ units (if $h > 0$) or left (if $h < 0$). Notice the sign is opposite to what you might expect.
• Reflection over the x-axis: $y = -f(x)$ flips the graph upside down.
• Reflection over the y-axis: $y = f(-x)$ flips the graph left to right.
• Vertical stretch/compression: $y = a \cdot f(x)$. If $|a| > 1$, the graph stretches away from the x-axis. If $0 < |a| < 1$, it compresses toward the x-axis.

A quick example: starting from $y = x^2$, the function $y = (x - 3)^2 + 2$ shifts the parabola 3 units right and 2 units up, placing the vertex at $(3, 2)$.

When multiple transformations are combined, apply horizontal shifts and reflections first (inside the function), then vertical stretches and shifts (outside the function).