Transformation Effects on Domain, Range, and Key Features

Why transformations change features (and how to track them fast)

In this chapter, you will focus on how transformations change domain, range, and key graph features such as intercepts, vertex/corner points, and asymptotes. The most reliable method is to track what happens to a few “anchor” features of the parent function (like intercepts, a vertex, or an asymptote) and then verify by evaluating a couple of sample points.

We will use side-by-side “before and after” comparisons for each function family. In each comparison, keep two ideas separate:

• Vertical changes (changes to outputs) mainly affect the range and y-locations of features.
• Horizontal changes (changes to inputs) affect the domain and x-locations of features.

1) Vertical changes: range changes, domain stays the same

Vertical transformations change the output values. If you transform a function into g(x)=a f(x)+k, then:

• Domain: stays the same as f (because the allowed inputs do not change).
• Range: changes because outputs are scaled by a and shifted by k.
• y-intercept: becomes g(0)=a f(0)+k (if 0 is in the domain).
• x-intercepts: generally change because solving g(x)=0 is different from solving f(x)=0 unless k=0 and a is nonzero.

Linear: before and after (vertical shift and vertical scale)

Confirm with sample points: For f, points (0,0), (1,1). For g, evaluate: g(0)=3, g(1)=5. The x-intercept comes from 2x+3=0 so x=-3/2.

Quadratic: before and after (vertical shift and vertical scale)

Confirm with sample points: g(0)=-6, g(1)=-3, g(2)=6. The smallest output occurs at x=0, so the range starts at -6.

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Absolute value: before and after (vertical shift)

Confirm with sample points: g(0)=4, g(2)=6, g(-2)=6.

Radical: before and after (vertical shift)

Confirm with sample points: g(0)=-5, g(4)=-3, g(9)=-2. Domain did not change because the inside is still x.

Practice set A (vertical changes)

For each, (i) predict domain and range, (ii) predict intercepts/vertex/corner if applicable, then (iii) confirm by evaluating the suggested sample points.

• A1. g(x)=x^2+7. Confirm with x=0,1,2.
• A2. g(x)=-2|x|+6. Confirm with x=0,1,3.
• A3. g(x)=\sqrt{x}+2. Confirm with x=0,1,9.
• A4. g(x)=5x-10. Confirm with x=0,2, and solve for the x-intercept.

2) Horizontal changes: domain changes and key x-locations move

Horizontal transformations change the input values. If you transform a function into g(x)=f(x-h), then the graph shifts right by h. If you use g(x)=f(x+h), it shifts left by h. The most important consequences:

• Domain: often changes (especially for radicals and reciprocals) because the allowed inputs are determined by what is inside f.
• Key x-locations: x-intercepts, vertex x-coordinate, corner x-coordinate, and vertical asymptotes shift horizontally.
• Range: usually stays the same for pure horizontal shifts (no vertical change), because outputs are the same values, just occurring at different x’s.

Quadratic: before and after (horizontal shift)

Confirm with sample points: Evaluate near the vertex: g(4)=0, g(3)=1, g(5)=1.

Absolute value: before and after (horizontal shift)

Confirm with sample points: g(-2)=0, g(-1)=1, g(-3)=1.

Radical: before and after (horizontal shift changes domain)

Confirm with sample points: g(5)=0, g(6)=1, g(9)=2. Notice x=0 is not allowed anymore because it would require \sqrt{-5}.

Reciprocal: before and after (horizontal shift moves vertical asymptote)

Confirm with sample points: g(4)=1, g(2)=-1, g(3) is undefined (matches the vertical asymptote at x=3).

Practice set B (horizontal changes)

• B1. g(x)=(x+1)^2. Predict vertex and range; confirm with x=-1,0,-2.
• B2. g(x)=|x-7|. Predict corner point; confirm with x=7,6,8.
• B3. g(x)=\sqrt{x+4}. Predict domain and starting point; confirm with x=-4,-3,0.
• B4. g(x)=1/(x+5). Predict domain and vertical asymptote; confirm with x=-4,-6,-5 (note which input is not allowed).

3) Reflections: effects on intercepts and symmetry

Reflections flip a graph across an axis (or the origin). They are especially noticeable in intercepts and symmetry.

• Reflect across the x-axis: g(x)=-f(x). Domain stays the same; range is reflected (outputs negated). The y-intercept changes sign: (0,f(0)) becomes (0,-f(0)). x-intercepts stay the same because -f(x)=0 happens exactly when f(x)=0.
• Reflect across the y-axis: g(x)=f(-x). Range stays the same; domain is mirrored (if the domain is not all real numbers). x-locations of features change sign: a point at (a,b) becomes (-a,b). The y-intercept stays the same because g(0)=f(0).
• Reflect through the origin: g(x)=-f(-x). Both flips happen: (a,b) becomes (-a,-b).

Linear: before and after (x-axis reflection)

Confirm with sample points: f(0)=-3 so g(0)=3. Solve f(x)=0 gives x=3/2; solve g(x)=0 gives the same.

Quadratic: before and after (x-axis reflection changes opening and range)

Confirm with sample points: f(1)=0, f(2)=1; then g(1)=0, g(2)=-1.

Absolute value: before and after (x-axis reflection keeps corner x-location)

Confirm with sample points: f(-3)=0, f(-2)=1; g(-3)=0, g(-2)=-1.

Radical: before and after (y-axis reflection changes domain)

Confirm with sample points: g(0)=0, g(-1)=1, g(1) is not allowed.

Reciprocal: before and after (y-axis reflection and intercept behavior)

Confirm with sample points: f(2)=1/2 while g(2)=-1/2; f(-2)=-1/2 while g(-2)=1/2.

Practice set C (reflections)

• C1. Let f(x)=x^2-4. Create g(x)=-f(x). Predict the x-intercepts of g and compare to f. Confirm by evaluating x=-2,0,2.
• C2. Let f(x)=|x-5|+1. Create g(x)=f(-x). Predict the new corner point. Confirm with x=-5,-4,-6.
• C3. Let f(x)=\sqrt{x-2}. Create g(x)=f(-x). Predict the domain of g. Confirm by testing x=-2,2,0.
• C4. Let f(x)=1/(x-1). Create g(x)=-f(x) and h(x)=f(-x). Predict how the asymptotes change for each. Confirm with one sample point on each side of the vertical asymptote.

4) Application: asymptote shifts for reciprocal-type functions

Reciprocal-type functions are a good place to practice transformation effects because their key features (vertical and horizontal asymptotes) are easy to track and strongly connected to domain and range restrictions.

Core model: g(x)=a/(x-h)+k

• Vertical asymptote: x=h (where the denominator is zero). This x-value is excluded from the domain.
• Horizontal asymptote: y=k (the output value the function approaches far left/right). This y-value is excluded from the range when the function is exactly of the form a/(x-h)+k with a≠0.
• Domain: all real numbers except x=h.
• Range: all real numbers except y=k.
• Intercepts: often change with h and k. The x-intercept (if it exists) comes from solving a/(x-h)+k=0.

Side-by-side comparison: parent vs shifted reciprocal

Intercept example (for g): To find the x-intercept, solve 1/(x-2)+3=0. Then 1/(x-2)=-3, so x-2=-1/3 and x=5/3. Notice how the horizontal shift and vertical shift both affect where (and whether) the graph crosses the x-axis.

Side-by-side comparison: negative scale and asymptote behavior

Multiplying by a changes the branch orientation and steepness, but it does not move the asymptotes when h=0 and k=0.

Practice set D (asymptote shifts and verification)

For each function, (i) state domain and range, (ii) identify vertical and horizontal asymptotes, (iii) predict whether x- and y-intercepts exist, then (iv) confirm by evaluating the listed sample points.

• D1. g(x)=4/(x+1). Check x=0 and x=-2. Identify the vertical asymptote and explain why that x-value is excluded from the domain.
• D2. g(x)=2/(x-5)-1. Check x=6 and x=4. Predict the range and verify that y=-1 is not reached.
• D3. g(x)=-3/(x+2)+4. Check x=-1 and x=-3. Solve for the x-intercept (if it exists) and compare it to the asymptote location.
• D4. g(x)=1/(x-2)+3. Confirm the asymptotes by evaluating values close to x=2 (for example x=1.9 and x=2.1) and far from it (for example x=100).