Transformations of functions Test

{"questions":[{"content":"Which equation shifts $y=f(x)$ left 2 and up 7?\u003Cbr\u003E[[choice-transformations-of-functions-test-1]]","widgets":{"choice-transformations-of-functions-test-1":{"type":"choice","options":["$y=f(x+2)+7$","$y=f(x-2)+7$","$y=f(x+7)+2$"],"answer":[0],"choice_feedback":["The input change x + 2 shifts left; the output addition shifts up.","An inside −2 shifts right rather than left. A left shift by 2 requires x + 2.","The 2 belongs inside f because it is the horizontal shift; the 7 belongs outside because it is the vertical shift."]}},"hints":[],"explanation":"The input change x + 2 shifts left; the output addition shifts up.","id":"transformations-of-functions-test-1","evidence_kind":"objective-concept-test","correct_feedback":"The input change x + 2 shifts left; the output addition shifts up.","feedback_visual":{"a":1,"b":1,"h":-2,"k":7,"formula":"f(x + 2) + 7","caption":"The chosen function is an example only. The same coordinate rule applies to any suitable f. Solid: original; dashed: transformed. The numbered points have the same original input.","base":[1,0,0],"bounds":[-7,7,-2,18],"type":"function_graph"}},{"content":"A point (3,−4) is on $y=f(x)$. Enter its new y-coordinate on $y=5-f(x)$. [[input-transformations-of-functions-test-2]]","widgets":{"input-transformations-of-functions-test-2":{"type":"input","answer":"9"}},"hints":[],"explanation":"Reflect the output and add 5: 5 − (−4) = 9.","id":"transformations-of-functions-test-2","evidence_kind":"objective-concept-test","correct_feedback":"Reflect the output and add 5: 5 − (−4) = 9.","incorrect_feedback":"The original output is −4. Subtracting that negative value from 5 increases the height rather than decreasing it."},{"content":"Which point on $y=f(3x-6)$ corresponds to (9,2) on $y=f(x)$?\u003Cbr\u003E[[choice-transformations-of-functions-test-3]]","widgets":{"choice-transformations-of-functions-test-3":{"type":"choice","options":["$(5,2)$","$(1,2)$","$(21,2)$"],"answer":[0],"choice_feedback":["Solve 3x − 6 = 9 to get x = 5.","Solving 3x − 6 = 9 gives x = 5. Factoring it as 3(x − 2) shows both the compression and the shift.","Apply the input expression in reverse to locate the new coordinate: solve 3x − 6 = 9 rather than applying 3(9) − 6."]}},"hints":[],"explanation":"Solve 3x − 6 = 9 to get x = 5.","id":"transformations-of-functions-test-3","evidence_kind":"objective-concept-test","correct_feedback":"Solve 3x − 6 = 9 to get x = 5.","deeper_explanation":"A transformed point must produce the old input 9 inside f. Solve $3x-6=9$ rather than applying that expression to 9. This reverses the input mapping.","feedback_visual":{"a":1,"b":3,"h":2,"k":0,"formula":"f(3x − 6)","caption":"The chosen function is an example only. The same coordinate rule applies to any suitable f. Solid: original; dashed: transformed. The numbered points have the same original input.","base":[1,0,0],"bounds":[-7,7,-4,12],"type":"function_graph"}},{"content":"The vertex of $y=f(x)$ is (−1,2). Find the vertex of $y=3f(x+4)-5$.\u003Cbr\u003E[[choice-transformations-of-functions-test-4]]","widgets":{"choice-transformations-of-functions-test-4":{"type":"choice","options":["$(-5,1)$","$(3,1)$","$(-5,-3)$"],"answer":[0],"choice_feedback":["The x-coordinate decreases by 4 and the output becomes 3(2) − 5.","Inside +4 shifts the old x-coordinate left by 4, so −1 becomes −5 rather than 3.","The old height 2 is first multiplied by 3 and then reduced by 5. Subtracting 5 alone omits the stretch."]}},"hints":[],"explanation":"The x-coordinate decreases by 4 and the output becomes 3(2) − 5.","id":"transformations-of-functions-test-4","evidence_kind":"objective-concept-test","correct_feedback":"The x-coordinate decreases by 4 and the output becomes 3(2) − 5.","feedback_visual":{"a":3,"b":1,"h":-4,"k":-5,"formula":"3f(x + 4) − 5","caption":"Example f(x) = (x + 1)² + 2 has the stated vertex (−1,2). The transformed vertex is (−5,1). Other suitable functions with that vertex obey the same point mapping.","base":[1,-1,2],"bounds":[-8,4,-2,14],"type":"function_graph"}},{"content":"A function has domain [2,8]. Find the domain of $g(x)=f(2x+2)$.\u003Cbr\u003E[[choice-transformations-of-functions-test-5]]","widgets":{"choice-transformations-of-functions-test-5":{"type":"choice","options":["$[0,3]$","$[2,8]$","$[6,18]$"],"answer":[0],"choice_feedback":["Solve 2 ≤ 2x + 2 ≤ 8 for x.","The composite accepts x only when 2x + 2 is in the old domain. The old interval cannot simply be kept.","This applies 2x + 2 to the old endpoints. Instead, solve for the new inputs that map into those endpoints."]}},"hints":[],"explanation":"Solve 2 ≤ 2x + 2 ≤ 8 for x.","id":"transformations-of-functions-test-5","evidence_kind":"objective-concept-test","correct_feedback":"Solve 2 ≤ 2x + 2 ≤ 8 for x.","deeper_explanation":"Both domain bounds constrain the inner expression: $2≤2x+2≤8$. Subtract 2 throughout and then divide by 2, giving $0≤x≤3$. This reverses the input map rather than transforming the old bounds forward."},{"content":"Why does $y=f(x/4)$ stretch horizontally by 4 rather than compress?\u003Cbr\u003E[[choice-transformations-of-functions-test-6]]","widgets":{"choice-transformations-of-functions-test-6":{"type":"choice","options":["A former input a is reached at x = 4a","The output is divided by 4","Every input becomes a/4 on the graph"],"answer":[0],"choice_feedback":["The coordinate mapping solves x/4 = a.","The output would be divided by 4 in f(x)/4. In f(x/4), it is the argument that changes.","Dividing the new x-coordinate by 4 must recover the old input a. Therefore the new coordinate is 4a, not a/4."]}},"hints":[],"explanation":"The coordinate mapping solves x/4 = a.","id":"transformations-of-functions-test-6","evidence_kind":"objective-concept-test","correct_feedback":"The coordinate mapping solves x/4 = a.","deeper_explanation":"Take any original input a. The equation $x/4=a$ gives $x=4a$, so distances from the y-axis are multiplied by 4. The corresponding output is unchanged.","feedback_visual":{"a":1,"b":0.25,"h":0,"k":0,"formula":"f(x/4)","caption":"The chosen function is an example only. The same coordinate rule applies to any suitable f. Solid: original; dashed: transformed. The numbered points have the same original input.","base":[1,0,0],"bounds":[-7,7,-4,12],"type":"function_graph"}},{"content":"A graph has range [−3,5]. Find the range of $y=1-2f(x)$.\u003Cbr\u003E[[choice-transformations-of-functions-test-7]]","widgets":{"choice-transformations-of-functions-test-7":{"type":"choice","options":["$[-9,7]$","$[-5,11]$","$[-3,5]$"],"answer":[0],"choice_feedback":["The negative multiplier reverses endpoint order: 1 − 2(5) = −9 and 1 − 2(−3) = 7.","The output is 1 − 2y, so the endpoint 5 maps to −9 and −3 maps to 7. Check both the minus sign and the added 1.","The changed outputs cannot retain the old range. The negative multiplier also reverses which endpoint is the minimum."]}},"hints":[],"explanation":"The negative multiplier reverses endpoint order: 1 − 2(5) = −9 and 1 − 2(−3) = 7.","id":"transformations-of-functions-test-7","evidence_kind":"objective-concept-test","correct_feedback":"The negative multiplier reverses endpoint order: 1 − 2(5) = −9 and 1 − 2(−3) = 7.","deeper_explanation":"The map $y\\mapsto1-2y$ is decreasing, so the largest old output becomes the smallest new output. Evaluate both endpoint images and put them in increasing order."},{"content":"A function has its unique maximum at (6,4). Find the unique minimum of $h(x)=-f(2-2x)+3$.\u003Cbr\u003E[[choice-transformations-of-functions-test-8]]","widgets":{"choice-transformations-of-functions-test-8":{"type":"choice","options":["$(-2,-1)$","$(2,-1)$","$(-2,7)$"],"answer":[0],"choice_feedback":["Solve 2 − 2x = 6 and transform the output to −4 + 3. The vertical reflection makes a maximum a minimum.","To reach the old maximum's input, solve 2 − 2x = 6. The negative coefficient gives x = −2, not 2.","The outside minus sign reflects the height 4 before adding 3. This makes the old maximum a minimum of height −1."]}},"hints":[],"explanation":"Solve 2 − 2x = 6 and transform the output to −4 + 3. The vertical reflection makes a maximum a minimum.","id":"transformations-of-functions-test-8","evidence_kind":"objective-concept-test","correct_feedback":"Solve 2 − 2x = 6 and transform the output to −4 + 3. The vertical reflection makes a maximum a minimum.","deeper_explanation":"The old maximum occurs at input 6, so solve $2-2x=6$ for its new location. The outside reflection changes the maximum into a minimum, and its height becomes $-4+3=-1$.","feedback_visual":{"a":-1,"b":-2,"h":1,"k":3,"formula":"−f(2 − 2x) + 3","caption":"Example f(x) = −(x − 6)² + 4 has the specified maximum. Its transformed graph has a minimum at (−2,−1). This illustrates the general point-mapping rule, not an assumption that every f is quadratic.","base":[-1,6,4],"bounds":[-5,9,-6,8],"type":"function_graph"}}],"mix":0,"tinkumo_module_mode":"test"}