Analyse reciprocal and square-reciprocal functions

What should be found first for the vertical asymptote?

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?

Reciprocal family features

A reciprocal family is described by its power, two asymptotes and signed scale. The asymptotes restrict the domain and range, while the power and scale determine the branch orientation

1. 1Find the vertical asymptote Set the denominator base to zero. The solution is the excluded input \(\displaystyle x-h=0\Rightarrow x=h\)
2. 2Read the horizontal asymptote The term outside the fraction gives the excluded output value \(\displaystyle f(x)=\frac{a}{\left(x-h\right)^p}+k\Rightarrow y=k\)
3. 3Exclude the asymptote values The vertical asymptote is excluded from the domain. The horizontal asymptote is excluded from the range \(\displaystyle D=\mathbb{R}\setminus\left\{h\right\},\quad R=\mathbb{R}\setminus\left\{k\right\}\text{ when }p=1\)
4. 4Use power parity and scale sign An odd reciprocal power gives opposite diagonal branches, while an even power puts both branches on the side set by the sign of the scale \(\displaystyle \frac{3}{x-1}+2\text{ has opposite branches};\quad\frac{3}{\left(x-1\right)^2}+2>2\)

Retained worked setup

\(\displaystyle f(x)=-12/(x-2)^2+5,\quad x=4\)

Calculate the denominator value at x=4.

Retained worked setup

\(\displaystyle f(x)=-12/(x-2)^2+5,\quad x=4\)

Use that denominator value to calculate f(4)

Translation sign in the denominator. One of these lines is right. Which one?

Build a full drawing from the existing features

Work on paper or a blank drawing surface. Use exact parent points, algebra, domain restrictions and symmetry; differentiation is not required. Label every allowed branch, intercept, endpoint and asymptote, then compare the complete drawing with the feedback

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 without calculus Substitute labelled points, use factors and signs between roots, compare symmetry and both ends, and use tables or simple inequalities to check the shape. Distinguish a local extremum from a bound on the whole domain. A plotted model is feedback, not your submitted drawing

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 or relation

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

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

Your complete graph construction

By hand, sketch the function shown 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

Review the whole drawing and its domain

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. The branches cannot join because x = −1 makes the denominator zero

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 or relation

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

Calculate the function value at x = 0.

Your complete graph construction

Sketch the function shown on its maximal domain. State the range, symmetry, asymptotes and intercepts. Explain whether the curve meets the x-axis

Review the whole drawing and its domain

Domain x≠2; range y>1; symmetry about x=2. Asymptotes x=2 and y=1. No x-intercept, because the function stays above 1; the y-intercept is shown below. The points (1,2) and (3,2) lie on the two branches, both tending to +∞ near 2.