Choose a representation
If f(p)=q, then f(x-h) reaches q at x=p+h. For -f(x)+k, keep the input and replace the output by -q+k. Inspect radicands and denominators before simplifying.
Worked example: follow the quantity
The graph of f contains (3,5). Define g(x)=f(x−8). What x-coordinate gives the corresponding point on g? Set the inner input x minus the shift equal to the original input. Solve that equation; the output stays unchanged. Substitute the given numbers: 3+8 = 11. Therefore the requested value is 11.
Contrast a plausible mistake
For that example, the tempting answer 24 follows this error: Multiply the input by the shift instead of adding. The requested quantity is 11. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
For real x, f(x)=√(3 x−15). What is the smallest permitted input? Require the radicand to be at least zero. Divide the resulting inequality by its positive coefficient. Substitute the given numbers: 15 ÷ 3 = 5. Therefore the requested value is 5.
Transfer before checking
The graph of f contains (4,6). Define g(x)=f(x−9). What x-coordinate gives the corresponding point on g? Try without viewing the answer. Check: 13. Set the inner input x minus the shift equal to the original input. Solve that equation; the output stays unchanged. Substitute the given numbers: 4+9 = 13.
Scope and conditions
An inside subtraction shifts the graph right. An outside subtraction shifts it down. A cancelled denominator factor can leave an excluded input in the original rule. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
The graph of f contains (3,5). Define g(x)=f(x−8). What x-coordinate gives the corresponding point on g?
- Set the inner input x minus the shift equal to the original input.
- Solve that equation; the output stays unchanged.
- Substitute the given numbers: 3+8 = 11.
- Therefore the requested value is 11.
Compare the other answer choices
Ignore the horizontal transformation.
Multiply the input by the shift instead of adding.
Shift left because the formula contains subtraction.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content