Parent Functions & Transformations — Challenge
A handful of basic shapes, moved and stretched, account for almost every graph you will meet.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Absolute value — |x| — distance from zero, always positive or zero.
- Open — Open excludes the endpoint (<, >); closed includes it (≤, ≥).
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Square root — For k ≥ 0, √k is the nonnegative number whose square is k.
- Factor — A number that divides evenly into another.
- Function — A rule pairing each input with exactly one output.
- Parabola — The U-shaped graph of a quadratic function.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
How does the graph of y=f(x)+2 differ from that of y=f(x)?
- Shifted up 2
- Shifted down 2
- Shifted right 2
- Shifted left 2
- Not shifted
Worked answer
Shifted up 2
Adding 2 to the output raises every point by 2.
So the graph moves up 2 units, keeping its shape.
Changes made outside the function affect the output and move the graph vertically.
Problem 2
How does the graph of y=f(x-5) differ from that of y=f(x)?
- Shifted left 5
- Shifted right 5
- Shifted up 5
- Shifted down 5
- Not shifted
Worked answer
Shifted right 5
To get the same output as before, x must now be 5 larger, since the function receives x-5.
So every feature of the graph appears 5 units further right.
Changes made to the input move the graph horizontally, and opposite to the sign.
Problem 3
The graph of y=-f(x) is a reflection of y=f(x) in what?
- The y-axis
- The x-axis
- The origin
- The line y=x
- It is not a reflection
Worked answer
The x-axis
Negating the output flips the sign of every y value while each x stays put.
Points above the axis move to the mirror position below it, so the reflection is in the x-axis.
y = f(-x) would negate the input and reflect in the y-axis instead.
Problem 4
The graph of y=f(2x) is which change to the graph of y=f(x)?
- A horizontal compression by factor 2
- A horizontal stretch by factor 2
- A shift right 2
- A shift up 2
- A reflection
Worked answer
A horizontal compression by factor 2
The function reaches any given input value at half the x it used to need, because the input is doubled before use.
So every feature arrives twice as soon, squeezing the graph towards the y-axis.
That is a horizontal compression by a factor of 2. Writing f(x2) would stretch it instead.
Problem 5
In y=2f(x)-1, what is the vertical shift?
- Up 1
- Down 1
- Right 1
- Left 1
- No shift
Worked answer
Down 1
The -1 is subtracted from the output after the stretching, so it moves the graph vertically.
Subtracting lowers every point by 1, a shift down 1.
The 2 is a separate vertical stretch, applied before the shift.
Problem 6
The graph of y=f(x) is shifted 3 units right and 2 units down. What is the equation of the new graph?
- y=f(x-3)-2
- y=f(x+3)-2
- y=f(x-3)+2
- y=f(x+3)+2
- y=f(x-2)-3
Worked answer
y=f(x-3)-2
A horizontal shift lives inside the bracket and works backwards: right 3 needs f(x-3). A vertical shift lives outside and is honest: down 2 needs -2. So y=f(x-3)-2.
Problem 7
What is the minimum value of y=|x-2|+|x+2|?
- 0
- 2
- 4
- 6
- 8
Worked answer
4
Read it as a distance: |x-2|+|x+2| is the total distance from x to 2 and to -2. For any x between -2 and 2, that total is exactly the gap between them, which is 4. Outside that stretch you are walking away from one point without getting closer to the other, so the total only grows. The minimum is 4.
Problem 8
The graph of y=x2 is reflected across the x-axis and then shifted up 5. Where does the new graph cross the x-axis?
- x=± 5
- x=±√5
- x=± 25
- Only at x=0
- It does not cross
Worked answer
x=±√5
Reflecting gives y=-x2; shifting up 5 gives y=-x2+5. Set y=0: x2=5. So x=±√5.