Apply and describe transformations of a function

What is the first calculation when mapping a point under this transformation?

Function

\(\displaystyle 2^{x-h}-2\)

Change h through all five settings and compare key points

Which description works for every displayed shift?

Transforming a function

A graph transformation can be tracked with one map for the input coordinate and one map for the output coordinate. Horizontal changes appear in the input of the rule in the reciprocal direction, while output changes stay outside the function

1. 1Map the coordinates separately Use the horizontal map for x and the vertical map for y. A negative vertical scale reflects the graph in the x-axis before any vertical translation \(\displaystyle x'=2x,\quad y'=-y\Rightarrow (x,y)\mapsto(2x,-y)\)
2. 2Reverse a horizontal translation inside the rule A graph shift right by t uses x-t inside the function \(\displaystyle x'=x+4\Rightarrow g(x)=f\left(x-4\right)\)
3. 3Use the reciprocal horizontal dilation factor A horizontal dilation by s maps x to sx, so the input of f is x divided by s. \(\displaystyle x'=3x\Rightarrow g(x)=f\left(\frac{x}{3}\right)\)
4. 4Preserve the order of combined changes A dilation and a translation on the same axis can give different results when their order is changed \(\displaystyle 2\left(x+1\right)=2x+2\ne2x+1\)

Given

\(\displaystyle x'=2(2)-3\)

Calculate the image x-coordinate of P=(2,-1)

Given

\(\displaystyle y'=-(-1)+4\)

Calculate the image y-coordinate of P=(2,-1)

Coordinate maps

\(\displaystyle x'=2x-3,\quad y'=-y+4\)

Swapping the coordinate maps. One of these lines is right. Which one?

From features to a complete hand sketch

Draw each complete labelled curve on paper or a blank drawing surface

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Lines and quadratics as complete graphs

The vertex is (1,3), the symmetry line is x=1 and the parabola opens downward. Its y-intercept is (0,1), paired by symmetry with (2,1). Solving f(x)=0 gives x=1±√(3/2). Domain R; range y≤3. Both ends go downward and there are no asymptotes

Rule

\(\displaystyle f(x)=-2(x-1)^2+3\)

Calculate the y-coordinate of the y-intercept of the worked parabola

Your complete hand-sketch task

On separate axes, draw the complete graphs y=3x−2 and y=(x+2)²−4. Label intercepts, domain and range, and the quadratic vertex and symmetry. Also explain why x²+4x and (x+2)²−4 give the same quadratic graph

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

The line passes through (0,−2) and (2/3,0), has gradient 3, and domain/range R. The quadratic has vertex (−2,−4), symmetry x=−2, intercepts (−4,0) and (0,0), domain R and range y≥−4. Expanding (x+2)²−4 gives x²+4x, a sum representation of the same function

Exponential key points and the omitted bound

Map the parent points (−1,1/2),(0,1),(1,2): they become (0,−3/2),(1,−1),(2,0). The curve increases, stays above y=−2 and approaches that horizontal asymptote to the left. Domain R; range (−2,∞). The asymptote is not an attained minimum

Rule

\(\displaystyle f(x)=2^{x-1}-2\)

Evaluate the function at x = 1.

Your complete hand-sketch task

Sketch g(x)=3^(−x)+1 by hand from parent key points. Label domain, range, intercepts and asymptote, and show the direction of growth/decay. Explain why it has no x-intercept

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

The curve decreases from unbounded positive values towards 1 as x increases. It has no x-intercept because it stays above 1

Logarithms preserve their input restriction

Require x+1>0, so the vertical asymptote is x=−1 and the domain is (−1,∞). Map parent points (1/e,−1),(1,0),(e,1) to (1/e−1,−2),(0,−1),(e−1,0). The curve increases, with range R. It falls without bound near x=−1 and grows without bound to the right

Rule

\(\displaystyle f(x)=\ln(x+1)-1\)

Find f(0) exactly

Your complete hand-sketch task

Sketch g(x)=log₁₀(x−2)+1. Use exact powers of ten to choose points, state domain/range, identify the vertical asymptote and find any intercepts. Explain why a y-intercept is unavailable

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Domain x>2, range R, vertical asymptote x=2. Points (21/10,0),(3,1),(12,2); x-intercept 21/10. There is no y-intercept because x=0 is outside the domain. The increasing curve approaches −∞ near 2 and is unbounded above as x increases

Sketch a whole periodic cycle

Use radians. Plot (0,1),(π/2,3),(π,1),(3π/2,−1),(2π,1) and join a smooth wave. Zeros in this interval are 7π/6 and 11π/6. The full periodic function has period 2π, domain R and range [−1,3]. Mark the included endpoints of the requested interval; the graph has no asymptotes

Rule

\(\displaystyle f(x)=2\sin x+1,\quad0\le x\le2\pi\)

Give the minimum value of the worked sine function

Your complete hand-sketch task

Sketch y=cos(2x)−1 on [−π,π]. Mark all maxima/minima and x-intercepts in the interval, the midline and included endpoints. State the period and range of the full function

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Period π; midline y=−1; range [−2,0]. Maxima and zeros at x=−π,0,π, each with y=0. Minima at x=−π/2 and π/2, each with y=−2. Complete two smooth cycles; do not join the key points with straight segments

Separate tangent branches at asymptotes

Mark vertical asymptotes x=−π/2 and x=π/2. The central branch passes through (0,−1), crosses the x-axis at π/4, and increases from −∞ to +∞. The range on this open interval is R. The full tangent function repeats with period π; a vertical asymptote is not part of the graph

Rule

\(\displaystyle f(x)=\tan x-1,\quad-\pi/2<x<\pi/2\)

Calculate the y-coordinate of the y-intercept of tan x − 1.

Your complete hand-sketch task

Sketch y=tan(2x)+1 on (−π/4,π/4). Label asymptotes and intercepts, and state its range and the period of the full function. Show which way the branch approaches each asymptote

1. 1Prepare the whole drawing State the domain and range. Label axes and scale; find intercepts, endpoints, asymptotes and exact key points before joining anything
2. 2Show the correct connections Draw every allowed branch with the correct direction and shape. Use open/closed endpoints, leave gaps at excluded inputs, and show end behaviour. A handful of correct dots is not a complete sketch
3. 3Check the mathematics Substitute labelled points, check signs between roots, and compare both ends of each branch. Use derivative/concavity information when available. A plotted model is feedback, not a drawing submitted by you
4. 4Written practice (ungraded) Compare your drawing with every listed feature and annotate corrections

Compare the complete drawing

Asymptotes x=±π/4; x-intercept −π/8; y-intercept 1; range R. The branch increases from −∞ to +∞. The full function has period π/2. The asymptotes remain excluded even if the displayed window cuts off the growing branches

Recover the map, then write the acting order

Assume an axis-aligned affine graph transformation with nonzero horizontal and vertical scales. Match the two original x-coordinates with their images to find the horizontal scale and shift; do the same with the y-coordinates. Solve for the original input to obtain the new function rule. The two given feature pairs determine this affine map, not an arbitrary function

1. 1Map the coordinates separately Use the horizontal map for x and the vertical map for y. A negative vertical scale reflects the graph in the x-axis before any vertical translation \(\displaystyle x'=2x,\quad y'=-y\Rightarrow (x,y)\mapsto(2x,-y)\)
2. 2Reverse a horizontal translation inside the rule A graph shift right by t uses x-t inside the function \(\displaystyle x'=x+4\Rightarrow g(x)=f\left(x-4\right)\)
3. 3Use the reciprocal horizontal dilation factor A horizontal dilation by s maps x to sx, so the input of f is x divided by s. \(\displaystyle x'=3x\Rightarrow g(x)=f\left(\frac{x}{3}\right)\)
4. 4Preserve the order of combined changes A dilation and a translation on the same axis can give different results when their order is changed \(\displaystyle 2\left(x+1\right)=2x+2\ne2x+1\)
5. 5Read the chronology The first named transformation acts on the original point before the second transformation acts on the result \(\displaystyle \mathbf{p}\xrightarrow{T_1}\mathbf{p}_1\xrightarrow{T_2}\mathbf{p}_2\)
6. 6State what counts as one transformation In these tasks, a translation by a vector (h,k) is one plane transformation, even when both components are nonzero. A dilation from a named axis and a reflection in a named axis are each separate transformations
7. 7Allow different valid sequences Two operations on different coordinates can commute. A translation can also be moved earlier if its vector is adjusted. Prove that the complete coordinate map is the same rather than rejecting an alternative merely because its order differs
8. 8Write your own sequence Write all three transformations in acting order, including each axis, factor and translation vector. Verify both labelled features and derive the whole transformed rule. Written practice is ungraded

Worked model: one complete sequence and two feature checks

For f(x)=x³−3x, the local maximum P=(−1,2) and local minimum Q=(1,−2) are to map to P′=(−1,−6) and Q′=(3,−2). The x-pairs give −a+h=−1 and a+h=3, hence a=2,h=1. The y-pairs give 2b+k=−6 and −2b+k=−2, hence b=−1,k=−4. One valid order is: 1. Dilate by factor 2 from the y-axis. 2. Reflect in the x-axis. 3. Translate by vector (1,−4). P follows (−1,2)→(−2,2)→(−2,−2)→(−1,−6); Q follows (1,−2)→(2,−2)→(2,2)→(3,−2). Reflection changes the maximum to a minimum and the minimum to a maximum. The first two operations may be exchanged because they act on different coordinates

1. 1Write your own sequence Write all three transformations in acting order, including each axis, factor and translation vector. Verify both labelled features and derive the whole transformed rule. Written practice is ungraded

Your three-transformation sequence from two stationary features

Use the same cubic f. Its maximum P=(−1,2) must map to P′=(−7,8), and its minimum Q=(1,−2) to Q′=(−1,0). Assume X=ax+h, Y=by+k with a>0 and b>0. Determine the four map constants. Write one sequence of exactly three plane transformations, stating both dilation axes/factors and the translation vector. Trace both points through every stage, derive the complete transformed function rule and explain the nature of both images. Then give a second valid order with the translation first and calculate its adjusted vector. Do not select from candidate sequences

1. 1Write your own sequence Write all three transformations in acting order, including each axis, factor and translation vector. Verify both labelled features and derive the whole transformed rule. Written practice is ungraded

Original and target features

\(\displaystyle (-1,2)\mapsto(-7,8),\quad(1,-2)\mapsto(-1,0);\qquad X=ax+h\)

After constructing your sequence, give the image x-coordinate of the original point (0,0)

Review the complete sequence, not just one coordinate

The map is X=3x−4, Y=2y+4. One valid order is: 1. Dilate by factor 3 from the y-axis. 2. Dilate by factor 2 from the x-axis. 3. Translate by vector (−4,4). P follows (−1,2)→(−3,2)→(−3,4)→(−7,8); Q follows (1,−2)→(3,−2)→(3,−4)→(−1,0). Positive vertical scale preserves maximum/minimum type. An alternative is translation left by four thirds and up by 2, then the two dilations in either order. Keeping (−4,4) when translating first would instead give X=3x−12 and Y=2y+8. Both original and transformed cubic have domain and range R.

1. 1Write your own sequence Write all three transformations in acting order, including each axis, factor and translation vector. Verify both labelled features and derive the whole transformed rule. Written practice is ungraded

Transfer the sequence to a reversed restricted graph

Now f is the restricted quadratic below. Map its vertex P=(1,−2) to P′=(−1,3), and its right endpoint Q=(3,2) to Q′=(−5,7), using X=ax+h and Y=y+k. Find a,h,k and write exactly three plane transformations: one positive dilation, one reflection and one translation, with their acting order and full details. Trace both endpoints, derive the complete rule, domain and range, and explain the new direction of travel. Check the image of the interior point (2,−1). Explain why reflecting the function rule is not permission to extend its original domain

1. 1Write your own sequence Write all three transformations in acting order, including each axis, factor and translation vector. Verify both labelled features and derive the whole transformed rule. Written practice is ungraded

Original and target features

\(\displaystyle (1,-2)\mapsto(-1,3),\quad(3,2)\mapsto(-5,7);\qquad X=ax+h\)

After writing the full sequence, give the image x-coordinate of the interior point (2,−1)

Review the endpoint order and restricted rule

The map is X=−2x+1, Y=y+5. One valid sequence is: 1. Dilate by factor 2 from the y-axis. 2. Reflect in the y-axis. 3. Translate by vector (1,5). P follows (1,−2)→(2,−2)→(−2,−2)→(−1,3); Q follows (3,2)→(6,2)→(−6,2)→(−5,7). The interior point maps to (−3,4), which satisfies the new rule. The original interval maps to [−5,−1], not [−1,−5]; both endpoints remain included. The range is [3,7]. As X increases from −5 to −1, the graph decreases from 7 to 3. The formula alone would permit other inputs, but they are not part of this transformed restricted function

1. 1Write your own sequence Write all three transformations in acting order, including each axis, factor and translation vector. Verify both labelled features and derive the whole transformed rule. Written practice is ungraded