Key takeaway: This algebra 2 transformations functions worksheet offers 22 printable problems on shifts, stretches, compressions, reflections, even and odd symmetry and root graphs. Standards F-BF.B.3 together with F-IF.C.7b apply. Problems 1 to 8 describe and write transformations, problems 9 to 11 test symmetry, problems 12 to 16 graph and read root functions, and problems 17 to 22 use real models. Set aside nearly an hour.
These 22 problems practice Common Core standards F-BF.B.3 and F-IF.C.7b. Describe and write shifts, stretches, compressions and reflections, test functions for even and odd symmetry, and graph square root and cube root functions.
The early problems are quick checks, the middle ones use graphs and tables, and the last six are word problems about motion, profit, packaging and growth.
Algebra 2 Transformations Functions Worksheet: problems 1–5
1. Describe the shift
Describe how each graph is related to the graph of \(y=x^{2}\).
- \(g(x)=x^{2}+5\)
- \(h(x)=x^{2}-3\)
- \(j(x)=(x-4)^{2}\)
- \(k(x)=(x+1)^{2}\)
2. Absolute value from a graph
The graph of an absolute value function is shown.
- Give the vertex.
- Write an equation of the form \(y=|x-h|+k\).
3. Transform a table
The table shows values of \(f(x)\). Make a table for each new function.
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
| \(f(x)\) | \(3\) | \(-1\) | \(2\) | \(5\) | \(0\) |
- \(f(x)+4\)
- \(f(x-1)\)
- \(2f(x)\)
- \(-f(x)\)
4. Reflections
Describe how each graph is related to the graph of \(y=\sqrt{x}\), and state the domain of each.
- \(g(x)=-\sqrt{x}\)
- \(h(x)=\sqrt{-x}\)
5. Stretch or compress
Describe how each graph is related to \(y=|x|\).
- \(g(x)=3|x|\)
- \(h(x)=\frac{1}{3}|x|\)
- \(k(x)=|3x|\)
Algebra 2 Transformations Functions Worksheet: problems 6–10
6. Write the equation
Write an equation for the graph after the transformation.
- \(y=x^{2}\) shifted left 3 and down 6
- \(y=|x|\) shifted right 5 and up 1
- \(y=\sqrt{x}\) shifted down 4
- \(y=2^{x}\) shifted up 3
7. Several transformations
Describe every transformation of \(y=x^{2}\) that produces \(g(x)=-2(x-1)^{2}+5\). State the vertex and whether the parabola opens up or down.
8. Exponential transformations
The graph of the parent function \(f(x)=2^{x}\) is shown. Let \(g(x)=2^{x}-3\) and \(h(x)=2^{x+1}\).
- Give the horizontal asymptote and the y-intercept of \(g\).
- Give the horizontal asymptote and the y-intercept of \(h\).
- Show that \(h(x)=2\cdot 2^{x}\).
9. Even, odd or neither
Use \(f(-x)\) to decide whether each function is even, odd or neither.
- \(f(x)=x^{4}-3x^{2}\)
- \(g(x)=x^{3}+x\)
- \(h(x)=x^{2}+x\)
10. Use symmetry
The function \(f\) is odd with \(f(1)=3\), \(f(2)=-5\) and \(f(3)=8\). The function \(g\) is even with \(g(2)=7\).
- Find \(f(-1)\), \(f(-2)\), \(f(-3)\) and \(f(0)\).
- Find \(g(-2)\).
Algebra 2 Transformations Functions Worksheet: problems 11–15
11. Parent functions and symmetry
Classify each parent function as even, odd or neither, and explain.
- \(y=|x|\)
- \(y=\sqrt[3]{x}\)
- \(y=\sqrt{x}\)
- \(y=2^{x}\)
12. Graph a square root function
Graph \(y=\sqrt{x-2}+1\). Then state the domain and range.
13. Graph a cube root function
Graph \(y=\sqrt[3]{x+1}-2\) by finding five points, one of them the center. State the domain and range.
14. Read a square root graph
The graph of a square root function is shown.
- State the domain and the range.
- Write the equation in the form \(y=a\sqrt{x-h}+k\).
15. Read a cube root graph
Write an equation of the form \(y=\sqrt[3]{x-h}+k\) for the graph.
Algebra 2 Transformations Functions Worksheet: problems 16–20
16. Move the points
The point \((6,-4)\) is on the graph of \(y=f(x)\). Find the corresponding point on each graph.
- \(y=f(x)+3\)
- \(y=f(x-2)\)
- \(y=2f(x)\)
- \(y=f(3x)\)
- \(y=f(-x)\)
- \(y=-f(x)\)
17. Ball toss
Noah tosses a ball straight up from the ground. Its height in feet after \(t\) seconds is \(h(t)=-16(t-2)^{2}+64\).
- Describe the transformations of \(y=t^{2}\).
- Find the maximum height and when it happens.
- Find \(h(0)\) and \(h(4)\), and tell how long the ball is in the air.
18. Weekly profit
The school store’s weekly profit in dollars from selling \(x\) hoodies is \(P(x)=-2(x-15)^{2}+450\).
- Find the maximum profit and the number of hoodies that produces it.
- Find \(P(10)\) and \(P(20)\). Why are they equal?
19. Gift box
Aisha packs a gift in a cube-shaped box with volume \(V\) cubic inches. The edge length of the outer shipping box is \(E(V)=\sqrt[3]{V}+2\) inches, which allows 1 inch of padding on each side.
- Describe how the graph of \(E\) is related to \(y=\sqrt[3]{V}\).
- Find \(E(27)\) and \(E(125)\).
- Find the gift volume when the shipping box edge is 9 inches.
20. Skid marks
Police estimate a car’s speed in miles per hour from the length \(d\) in feet of its skid marks with \(s(d)=\sqrt{24d}\).
- Find the speed for skid marks of 6 feet, 54 feet and 150 feet.
- Describe \(s\) as a transformation of \(y=\sqrt{d}\).
- How long are the skid marks of a car going 48 mph?
Algebra 2 Transformations Functions Worksheet: problems 21–22
21. Bacteria colony
Mei’s science project grows a colony of \(N(t)=500\cdot 2^{t}\) bacteria after \(t\) hours.
- Find \(N(0)\) and \(N(3)\).
- Write \(N(t+2)\) and show that it equals \(4N(t)\).
- Explain in words why shifting the graph left 2 hours gives the same graph as stretching it vertically by a factor of 4.
22. Thermostat error
Liam’s thermostat is set to 68°F. The error when the room temperature is \(T\) degrees is \(E(T)=|T-68|\).
- Describe how the graph of \(E\) is related to \(y=|T|\).
- Find \(E(65)\) and \(E(71)\).
- At which temperatures is the error exactly 5 degrees?
How do the first eight problems work?
Problem 1 compares \(x^{2}+5\), \(x^{2}-3\) and \((x-4)^{2}\) with \(y=x^{2}\). Reading a V-shaped graph in problem 2 gives the vertex and an equation. Then problem 4 reflects \(\sqrt{x}\), so \(-\sqrt{x}\) and \(\sqrt{-x}\) have different domains. In problem 5, \(3|x|\) stretches vertically, while \(|3x|\) compresses horizontally, and the two look different from each other.
Where do students slip with symmetry and points?
Many treat \(f(-x)\) as a sign flip on the whole expression. Instead, substitute \(-x\) for every \(x\). In problem 9, \(x^{4}-3x^{2}\) is even and \(x^{3}+x\) is odd. Problem 10 uses an odd function with \(f(1)=3\), so \(f(-1)=-3\). Problem 16 moves the point \((6,-4)\); the output changes with outside edits and the input changes with inside edits.
What do the modeling problems add?
The weekly profit \(P(x)=-2(x-15)^{2}+450\) has its vertex at 15 hoodies, where profit peaks at 450 dollars. The bacteria colony \(500\cdot 2^{t}\) starts at 500. The thermostat error \(|T-68|\) bottoms out at zero when the room hits 68 degrees. Finally, the skid mark problem uses a square root model, so a longer skid means a higher speed, but with slowing growth.
How can you check your work?
Pick any transformed graph you drew and verify two points by substituting into the new equation. If the points land on your curve, the shift directions are right. Afterward, use the key and try the quiz to confirm.
See every chapter on the Algebra 2 math page.
