2.2 Graphs of Functions
Ideas in this lesson
- Connect graph points, values, and tables
- Draw lines, constants, and restricted segments
- Draw quadratic and higher-power graphs
- Draw square-root and odd-root graphs
- Draw reciprocal graphs without false connections
- Graph absolute values and sign rules
- Select and justify graphing windows
- Construct piecewise graphs correctly
- Graph floor functions and step boundaries
- Use the vertical-line test and read finite domains
- Determine whether implicit equations define a function
- Recover formulas from segments and semicircles
- Compare power, root, shift, and scale families
- Build and graph contextual cost and physical models
- Investigate inverse-power relations and compressed steps
- Construct and explain graphs of absolute outputs
Connect graph points, values, and tables · Worked example
For \(f(x)=x^{2} - 4\), make a five-row table at x=−2,−1,0,1,2; plot the points and sketch the full curve. What point corresponds to input 3? Write or draw your full response before checking the answer. A teacher must review the construction or argument. Complete the written work or drawing first. The selected answer checks the result only; your reasoning or drawing needs teacher/self-review.
Answer: Formula \(y=x^{2} - 4\) on all real x; points \((-2,0)\), \((-1,-3)\), \((0,-4)\), \((1,-3)\), \((2,0)\). The point for input 3 is (3,5).
The table outputs are [0, -3, -4, -3, 0].
Plot input horizontally and output vertically.
The full graph is an upward parabola with vertex (0,−4); f(3)=5.
Written-work review:
Plot input horizontally and output vertically.
The full graph is an upward parabola with vertex (0,−4); f(3)=5.
Written-work review:
- Correct formula or tabulated coordinates.
- Labeled axes with a sensible scale; correct domain restrictions.
- Correct shape, key points, and open/closed endpoints; no spurious joining across gaps.