Math101 · Independent Ontario learning library
Math101learn.math101.caGraphs of Functions
A rigorous guide to reading, sketching, and verifying function graphs through features and transformations.
Precise definition
The graph of $f$ is the set $\{(x,f(x)):x\in\operatorname{dom}f\}$. Key features include intercepts, domain, range, intervals of increase/decrease, extrema, symmetry, continuity, asymptotes, concavity, and end behaviour. A graph is a representation of a rule, not the function's full definition by itself.
Notation and mathematical language
Transformations of $y=f(x)$ include $f(x-h)+k$ (right $h$, up $k$), $af(x)$ (vertical scaling/reflection), and $f(bx)$ (horizontal scale by $1/|b|$ and reflection if $b<0$). Inside transformations act inversely on input coordinates.
Conceptual picture
Parent-function features move predictably, while algebra identifies exact intercepts and restrictions. Tables sample points but do not prove behaviour between them. A graphing window can conceal asymptotes or create apparent intersections.
Conditions and key results
The vertical-line test identifies whether a plotted relation defines $y$ as a function of $x$. Holes retain original domain exclusions after cancellation. Numerical plots approximate curves and can miss rapid change, repeated roots, or close features.
A reliable strategy
- Determine domain, symmetry, intercepts, discontinuities, asymptotes, and end behaviour algebraically.
- Start from a parent graph and apply transformations in a tracked coordinate order.
- Plot enough exact landmarks and use rate or derivative information when available.
- Verify the sketch against the formula outside one calculator window.
Fully worked example
Interpretation and application
Graphs model costs, trajectories, populations, and data trends. An exact formula defines a graph; a fitted graph summarizes data approximately. Neither graphical association nor visual overlap alone proves causation.
Common mistakes
Verification and reasonableness
- Substitute intercepts and transformed landmark coordinates.
- Compare end behaviour with leading terms or asymptotic analysis.
- Change the plotting window and pair technology with exact algebra.
Practice
- Find the vertex of $(x+2)^2-5$.
- What is the range of $-x^2+4$?
- What test checks whether a relation is a function of $x$?
Answers and brief solutions
- $(-2,-5)$.
- $(-\infty,4]$.
- The vertical-line test.
Further deduction
For a rational function, factor first but retain restrictions. $(x^2-1)/(x-1)$ equals $x+1$ only for $x\ne1$, so its graph is the line $y=x+1$ with a hole at $(1,2)$. The simplified formula alone loses that feature; domain is part of graph identity.
Multiplicity connects an algebraic factor to local graph shape. If $f(x)=(x-1)^2(x+2)$, the zero $x=1$ has even multiplicity, so the sign does not change and the graph touches the axis there; $x=-2$ has odd multiplicity one, so the graph crosses. End behaviour comes from the leading term $x^3$: it falls to the left and rises to the right. Symmetry provides another audit: $f(-x)=f(x)$ identifies an even function symmetric about the vertical axis, while $f(-x)=-f(x)$ identifies an odd function symmetric about the origin. These tests cannot replace plotting all important features, but they constrain any plausible sketch. A viewing window that hides a turning point or intercept does not override factor, parity, domain, and asymptotic reasoning; the symbolic structure remains the primary evidence.
Related topics
Explore the idea
Function transformation
Change one quantity at a time and connect what moves to Graphs of Functions.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
What is the vertex x-coordinate of $3(x-5)^2+1$?
- Compare with $a(x-h)^2+k$.
- $h=5$, so the vertex x-coordinate is 5.
End of lesson
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