Analyse the general rational-power family

What should be done first with a rational exponent p/q?

For the rule three divided by x minus h, plus two, the vertical asymptote is at x = h. Change h through all five settings. The parent inputs stay −2, −1, 1 and 2; compare their mapped coordinates

What changes when h increases by 1?

Rational-power families

Reduce the rational exponent first, then use the reduced numerator and denominator to determine the domain and broad shape. Transformations map the parent features to the new endpoint or centre and key points

1. 1Reduce the exponent Write p/q in lowest terms before using denominator parity or classifying the family \(\displaystyle \frac{8}{12}=\frac{2}{3}\)
2. 2Use denominator parity for the domain An even denominator requires a non-negative radicand, while an odd denominator permits every real radicand \(\displaystyle q=4\Rightarrow u\ge0,\quad q=3\Rightarrow u\in\mathbb{R}\)
3. 3Compare p and q for shape The reduced index p/q is above one for a polynomial-like shape and below one for a root-like shape \(\displaystyle p=5>q=2\Rightarrow\text{polynomial-like}\)
4. 4Map parent features Use u = n(x − h) to map exact parent points, with the endpoint or centre at (h, k) \(\displaystyle u=1\Rightarrow\left(x,y\right)=\left(h+\frac{1}{n},k+a\right),\quad f(h)=k\)

Given

\(\displaystyle u=9-1\)

First find the parent input u at x = 9.

Given

\(\displaystyle f(9)=2\left(8\right)^{\frac{2}{3}}-5=2(4)-5\)

Now evaluate the transformed function at x = 9.

Using an unreduced exponent. One of these lines is right. Which one?

From features to a complete hand sketch

Draw each complete labelled curve on paper or a blank drawing surface

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Two reciprocal branches

Domain: all real x except 1; range: all real y except 2. Draw dashed asymptotes x = 1 and y = 2. The intercepts are (−1/2,0) and (0,−1). On the left branch y < 2 and the curve falls from near 2 to −∞ as x approaches 1. On the right branch y > 2 and it falls from +∞ towards 2. Join neither branch across x = 1.

Rule

\(\displaystyle f(x)=\frac{3}{x-1}+2\)

Use the rule to find the y-coordinate when x = 2.

Your complete hand-sketch task

By hand, sketch g(x) = 4/(x+1)−2 on its maximal domain. Give both intercepts, both asymptotes and the behaviour on each side of the excluded input. Include exact points on both branches and explain why they cannot join

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Domain R except −1; range R except −2. Asymptotes x=−1 and y=−2. Intercepts (1,0) and (0,2); other useful points (−2,−6) and (3,−1). Both branches decrease; the left lies below −2 and tends to −∞ at −1 from the left, while the right tends to +∞ at −1 from the right

A square reciprocal has different branch signs

The denominator is positive wherever defined, so both branches lie above y = −1. Domain: x ≠ −1; range: y > −1. The graph is symmetric about x = −1. Both sides tend to +∞ near the vertical asymptote; far away they approach −1. Intercepts: x = −1 ± √2 and y = 1.

Rule

\(\displaystyle f(x)=\frac{2}{(x+1)^2}-1\)

Calculate the function value at x = 0.

Your complete hand-sketch task

Sketch g(x)=1/(x−2)²+1 on its maximal domain. State the range, symmetry, asymptotes and intercepts. Explain whether the curve meets the x-axis

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Domain x≠2; range y>1; symmetry about x=2. Asymptotes x=2 and y=1. No x-intercept, because 1/(x−2)²+1>1; y-intercept (0,5/4). The points (1,2) and (3,2) lie on the two branches, both tending to +∞ near 2.

An endpoint is not an asymptote

The radicand requires x ≥ −1. Begin with the filled endpoint (−1,3). Exact points are (0,1), (3,−1) and (8,−3). The curve decreases, becoming less steep, and has range y ≤ 3. Its x-intercept is (5/4,0). It has no vertical or horizontal asymptote; do not extend it left of its endpoint

Rule

\(\displaystyle f(x)=3-2\sqrt{x+1}\)

Find the exact x-intercept by solving 3 − 2√(x+1) = 0.

Your complete hand-sketch task

Sketch g(x)=√(2−x)−1. State the maximal domain and range, mark its endpoint and x-intercept, and label two further exact points. Explain which direction the branch extends

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Domain x≤2; range y≥−1. Filled endpoint (2,−1), x-intercept (1,0), points (−2,1) and (−7,2); y-intercept (0,√2−1). The branch extends left and upward. It has no asymptote; joining a second reflected branch would change the function

Odd roots and a vertical tangent

The real cube root allows every real input. Plot (−7,0), (0,1), the centre (1,2), (2,3) and (9,4). Join a continuous increasing curve with a vertical tangent at the centre, not a gap. Domain and range are R. The two sides have different concavity and there are no vertical or horizontal asymptotes

Rule

\(\displaystyle f(x)=\sqrt[3]{x-1}+2\)

Find f(−7) using the real cube root

Your complete hand-sketch task

Sketch this rule using the real cube root followed by squaring. Label the cusp, domain, range, intercepts and symmetry. Compare its left branch with the increasing cube-root model

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Domain R; range y≥−2; cusp (−1,−2) and symmetry about x=−1. Points (−2,−1),(0,−1),(−9,2),(7,2). The x-intercepts are −1±2√2. The graph decreases to the cusp and then increases; the cusp is not an asymptote or a differentiable turning point

Extend the parity pattern beyond degree four

The key points (−1,−1),(0,0),(1,1) lie on every positive odd integer power. For x^5, join one increasing curve, flatter near zero and steeper away from it, with origin symmetry and a stationary inflection at zero. Domain and range are R. For a positive even power the two ends instead rise, the graph is symmetric about the y-axis and its range is [0,∞). Neither family has a vertical or horizontal asymptote

Rule

\(\displaystyle f(x)=x^5\)

Evaluate the fifth power at x = 2.

Your complete hand-sketch task

Sketch g(x)=−(x−1)^6+2, giving exact key points, intercepts, symmetry, range and end behaviour. Then describe what changes for −(x−1)^n+2 when the positive integer n is odd instead of even. Treat n=1 separately when discussing a horizontal tangent

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

For n=6, the maximum is (1,2), symmetry line x=1, points (0,1),(2,1), and x-intercepts 1±2^(1/6). Domain R, range y≤2 and both tails fall. For every positive even n the same range and symmetry hold. For odd n, domain/range are R, the graph decreases through centre (1,2) and has opposite end directions. Odd n≥3 gives a stationary inflection; n=1 gives a straight line with slope −1.