Investigate related function graphs

A point (2,3) belongs to a base graph. The graph moves one unit right and two units down. Where does that point go?

The dashed curve is y=1/x; the solid curve is y=a/(x−h)+k. Change one parameter at a time. Try a negative and fractional a, then both signs of h and k. Move u to compare P with Q. Scales stay fixed. Dashed straight lines mark transformed asymptotes

Hold a, h and u fixed. If k increases by one, what happens to Q?

One mapping, different graph families

Write the base rule as f. In g(x)=a f(x−h)+k, the letter f tells you to evaluate that base rule. A point (u,v) on the base graph moves to (u+h,av+k)

1. 1Horizontal shift Replacing x with x−h moves the graph right by h. Negative h means left
2. 2Output changes Multiply the old output by a, then add k. Negative a reflects in the horizontal x-axis before the vertical translation
3. 3Size change Absolute a greater than one stretches vertically; between zero and one compresses vertically. Keep a nonzero in these investigations
4. 4Domain and special features Carry each allowed input across by h. A square-root endpoint is an actual point; a reciprocal asymptote is an excluded line
5. 5Test a pattern Hold all but one parameter fixed. Record several points before and after the change, compare with the rule and test an unused setting

Rule or mapping

\(\displaystyle y_Q=-\frac12(4)+3\)

For P=(−1,4), use a=−1/2, h=2 and k=3. Find Q’s y-coordinate

Why is x−h inside the rule even though points move right by h?

Reciprocal family

Read the base shape first, then connect the transformed rule to its coordinates

1. 1Excluded input Division by zero is undefined. The transformed graph has no point at x=h.
2. 2Asymptotes The vertical line x=h and horizontal line y=k are approached. In this family with nonzero a, neither line is crossed
3. 3Two branches Keep the branches separate. A steep line across the gap would be a drawing error, not part of this function

Rule or mapping

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

Evaluate the reciprocal transformation at x=4.

Square root family

Read the base shape first, then connect the transformed rule to its coordinates

1. 1Principal root The base output √x is nonnegative, so the base graph begins at (0,0) and is defined only for x≥0.
2. 2Endpoint and domain The endpoint moves to (h,k). The allowed inputs are x≥h, even when a is negative
3. 3Output direction For positive a the outputs are at least k. For negative a the outputs are at most k. The endpoint is included

The dashed curve is y=√x; the solid curve is y=a√(x−h)+k. Change one parameter at a time. Try a negative and fractional a, then both signs of h and k. Move u to compare P with Q. Scales stay fixed. E marks the included endpoint

If only a changes from positive to negative, which input restriction remains?

Rule or mapping

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

Evaluate the square-root transformation at x=6.

Cubic family

Read the base shape first, then connect the transformed rule to its coordinates

1. 1Negative inputs Cubing preserves the input sign: for example, (−2)³=−8.
2. 2Centre The base graph passes through (0,0). Its centre moves to (h,k), but this is not a maximum or minimum
3. 3Both directions Every real input is allowed. With nonzero a, outputs are unbounded in both directions; negative a reverses increasing to decreasing

The dashed curve is y=x³; the solid curve is y=a(x−h)³+k. Change one parameter at a time. Try a negative and fractional a, then both signs of h and k. Move u to compare P with Q. Scales stay fixed

What kind of point is (h,k) for this cubic family?

Rule or mapping

\(\displaystyle g(x)=-\frac12(x-2)^3+1\)

Evaluate this cubic transformation at x=0.

Exponential growth family

Read the base shape first, then connect the transformed rule to its coordinates

1. 1Variable exponent The input is the exponent. Increasing a base input by one doubles the base output; it does not add a fixed amount
2. 2Reference point The base passes through (0,1), so the transformed graph passes through (h,a+k). It need not pass through (h,k)
3. 3Horizontal asymptote The base output is always positive. The transformed graph approaches y=k without reaching it. Negative a reverses its increasing direction

Dashed curve

\(\displaystyle y=2^x\)

Solid curve

\(\displaystyle y=a\times 2^{x-h}+k\)

Change one parameter at a time. Try a negative and fractional a, then both signs of h and k. Move u to compare P with Q. Scales stay fixed. Dashed straight lines mark transformed asymptotes

What happens to the base graph’s increasing direction when a becomes negative?

Rule or mapping

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

Evaluate this exponential transformation at x=0.

Exponential decay family

Read the base shape first, then connect the transformed rule to its coordinates

1. 1Halving Increasing the base input by one halves its output. Negative inputs give larger positive outputs, not negative outputs
2. 2Both exponential bases The base (1/2)^x equals 2^(−x), a reflection of 2^x in the vertical y-axis
3. 3Distinguish reflections Changing a to negative reflects outputs in the horizontal x-axis. It does not merely replace base 2 by base 1/2. After shifting, the horizontal asymptote is y=k.

Dashed curve

\(\displaystyle y=\left(\frac12\right)^x\)

Solid curve

\(\displaystyle y=a\times\left(\frac12\right)^{x-h}+k\)

Change one parameter at a time. Try a negative and fractional a, then both signs of h and k. Move u to compare P with Q. Scales stay fixed. Dashed straight lines mark transformed asymptotes

With a negative, what is the transformed decay graph’s direction?

Rule or mapping

\(\displaystyle g(x)=-2\left(\frac12\right)^{x-2}+1\)

Evaluate the decay transformation at x=1.

A student doubles the output and then shifts up three, but writes 2(f(x)+3). What should they write?

Investigate and report

Choose two different base families. Make a prediction, run the controls systematically and check it with coordinates

1. 1Design Vary one parameter at a time through at least three settings, keeping the other two fixed. When varying a, include a negative and a fractional value. Record each setting and three corresponding points
2. 2Test Explain your pattern using the coordinate mapping, then test an unused setting. Check any endpoint or excluded input separately \(\displaystyle (u,v)\mapsto(u+h,av+k)\)
3. 3Report Include labelled graphs with the same scales, your table, a general claim and an algebraic explanation. State what the finite tests support and why the mapping justifies the claim
4. 4Review Ask a teacher or peer to check axes, scales, domains, point mapping and reasoning. A few correct numeric responses do not certify an independent investigation

Which evidence best supports a general transformation claim?