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Transformations of Functions — Easy Worksheet 2
Grade 11 · MathSheets Name: ____________________ Date: __________
Use the rules for shifts, stretches and reflections.
How to solve
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Example
The graph of y = |x| is shifted 2 units right. Write the equation of the new graph.
Answer: y = |x - 2|
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = |x|. Shifting 6 left makes h = −6, so x is replaced by x + 6. Shifting 6 up makes k = 6, so add 6 at the end. So y = |x + 6| + 6.
✓ Correct!
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = |x|. Shifting 6 left makes h = −6, so x is replaced by x + 6. Shifting 6 up makes k = 6, so add 6 at the end. So y = |x + 6| + 6.
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = x² is shifted 5 units left, then shifted 1 unit up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The vertex of the V (its lowest point) is at (4, −4), so h = 4 and k = −4. It opens upward like y = |x|, so a = 1. Use y = a·f(x − h) + k: y = |x − 4| − 4. Check: one unit right of the vertex, at x = 5, the graph is at y = −3.
✓ Correct!
Not quite. The answer is
💡 The vertex of the V (its lowest point) is at (4, −4), so h = 4 and k = −4. It opens upward like y = |x|, so a = 1. Use y = a·f(x − h) + k: y = |x − 4| − 4. Check: one unit right of the vertex, at x = 5, the graph is at y = −3.
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = x² is reflected across the x-axis, shifted 3 units left, then shifted 1 unit up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The inside f(x + 3) shifts the graph 3 left, so add −3 to x: −3 + (−3) = −6. The outside adds 5 to y: −2 + 5 = 3. The new point is (−6, 3).
✓ Correct!
Not quite. The answer is
💡 The inside f(x + 3) shifts the graph 3 left, so add −3 to x: −3 + (−3) = −6. The outside adds 5 to y: −2 + 5 = 3. The new point is (−6, 3).
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = x² is shifted 1 unit left, then shifted 3 units up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = x³. The reflection across the x-axis makes a = −1 (multiply the whole function by −1). Shifting 3 right makes h = 3, so x is replaced by x − 3. Shifting 3 up makes k = 3, so add 3 at the end. So y = −(x − 3)³ + 3.
✓ Correct!
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = x³. The reflection across the x-axis makes a = −1 (multiply the whole function by −1). Shifting 3 right makes h = 3, so x is replaced by x − 3. Shifting 3 up makes k = 3, so add 3 at the end. So y = −(x − 3)³ + 3.
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = x² is reflected across the x-axis, shifted 3 units left, then shifted 3 units up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = x². The reflection across the x-axis makes a = −1 (multiply the whole function by −1). Shifting 5 right makes h = 5, so x is replaced by x − 5. Shifting 1 down makes k = −1, so subtract 1 at the end. So y = −(x − 5)² − 1.
✓ Correct!
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = x². The reflection across the x-axis makes a = −1 (multiply the whole function by −1). Shifting 5 right makes h = 5, so x is replaced by x − 5. Shifting 1 down makes k = −1, so subtract 1 at the end. So y = −(x − 5)² − 1.
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = x³ is reflected across the x-axis, shifted 4 units right, then shifted 1 unit up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The vertex of the V (its lowest point) is at (−1, 1), so h = −1 and k = 1. It opens upward like y = |x|, so a = 1. Use y = a·f(x − h) + k: y = |x + 1| + 1. Check: one unit right of the vertex, at x = 0, the graph is at y = 2.
✓ Correct!
Not quite. The answer is
💡 The vertex of the V (its lowest point) is at (−1, 1), so h = −1 and k = 1. It opens upward like y = |x|, so a = 1. Use y = a·f(x − h) + k: y = |x + 1| + 1. Check: one unit right of the vertex, at x = 0, the graph is at y = 2.
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = |x| is shifted 5 units left, then shifted 2 units up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The inside f(x − 5) shifts the graph 5 right, so add 5 to x: −1 + 5 = 4. The outside adds 4 to y: −1 + 4 = 3. The new point is (4, 3).
✓ Correct!
Not quite. The answer is
💡 The inside f(x − 5) shifts the graph 5 right, so add 5 to x: −1 + 5 = 4. The outside adds 4 to y: −1 + 4 = 3. The new point is (4, 3).
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = |x| is reflected across the x-axis, shifted 3 units right, then shifted 3 units down. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = x². The reflection across the x-axis makes a = −1 (multiply the whole function by −1). Shifting 3 left makes h = −3, so x is replaced by x + 3. Shifting 4 up makes k = 4, so add 4 at the end. So y = −(x + 3)² + 4.
✓ Correct!
Not quite. The answer is
💡 Use y = a·f(x − h) + k with f(x) = x². The reflection across the x-axis makes a = −1 (multiply the whole function by −1). Shifting 3 left makes h = −3, so x is replaced by x + 3. Shifting 4 up makes k = 4, so add 4 at the end. So y = −(x + 3)² + 4.
💡 How to solve: Transformations of Functions
For g(x) = a·f(x - h) + k: h moves the graph right (h > 0) or left, k moves it up or down, |a| > 1 stretches it vertically, and a negative a reflects it across the x-axis. A point (x, y) on f moves to (x + h, a·y + k) on g.
Try a similar one
The graph of y = x³ is shifted 6 units left, then shifted 1 unit up. Write the equation of the new graph.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Transformations of Functions — Easy Worksheet 2
Use the rules for shifts, stretches and reflections.