Transformations of Functions: Shifts in x and y Using Function Notation

1) Vertical Shifts: f(x)+k Changes Outputs

A vertical shift changes the output of a function without changing the input. If the original function is y=f(x), then the transformed function

y = f(x) + k

adds the constant k to every output value. This moves the graph straight up if k>0 and straight down if k<0.

Effect on outputs (function values)

For any input x:

• Original output: f(x)
• New output: f(x)+k

So the change in output is always the same: +k.

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Effect on range

If the range of f is some set of outputs, then the range of f(x)+k is that entire set shifted by k. In interval language:

• If range(f) = [a, b], then range(f+k) = [a+k, b+k].
• If range(f) = [a, \infty), then range(f+k) = [a+k, \infty).

Example (range shift): Suppose a function has minimum output 2 (so its range is [2, \infty)). Then f(x)-5 has minimum output 2-5=-3, so its range becomes [-3, \infty).

Quick check with a point

If a point (x, y) is on y=f(x), then after a vertical shift by k, the new point is:

(x, y) \rightarrow (x, y+k)

2) Horizontal Shifts: f(x-h) Changes Inputs

A horizontal shift changes the input to the function. If the original function is y=f(x), then the transformed function

y = f(x-h)

replaces x with x-h. This moves the graph horizontally, but the direction can feel “backwards” because the shift is happening inside the function.

Why the graph moves opposite the sign

To see the direction, compare inputs that produce the same output. Suppose f(a)=b. In the transformed function:

f(x-h)=b happens when x-h=a, so x=a+h.

That means the output b that used to occur at input a now occurs at input a+h. So every feature moves right by h.

• f(x-h) shifts the graph right by h.
• f(x+h) shifts the graph left by h.

Point mapping for horizontal shifts

If a point (x, y) is on y=f(x), then after a horizontal shift by h to the right (using f(x-h)), the new point is:

(x, y) \rightarrow (x+h, y)

Notice: the y-value stays the same; only the x-value changes.

3) Finding New Key Points Using Point Mapping (Instead of Re-Plotting)

When sketching a transformed graph, you often do not need to recompute lots of values. Instead, take key points from the original graph (intercepts, vertex, corners) and map them to new locations.

Core mapping rules

Start with a known point on y=f(x): (x, y).

Combining a horizontal and vertical shift

For y = f(x-h) + k, apply both moves:

(x, y) \rightarrow (x+h, y+k)

Step-by-step sketching method (recommended):

• Pick 3–5 key points on the original graph of f.
• Map each point using the appropriate rule(s).
• Plot the new points.
• Draw the same shape through the new points (same curvature/corners), just shifted.

Mini-example with mapping

Suppose f has known points (-1, 2), (0, 0), and (2, 3). Sketch g(x)=f(x-4)-1.

• Right 4: x increases by 4.
• Down 1: y decreases by 1.

Map each point:

• (-1, 2) \rightarrow (3, 1)
• (0, 0) \rightarrow (4, -1)
• (2, 3) \rightarrow (6, 2)

Plot (3,1), (4,-1), (6,2), then draw the same overall shape as f, shifted.

4) Applying Shifts Consistently Across Function Families

The notation rules do not change with the family. Whether the base function is linear, quadratic, or absolute value, the same shift logic applies: +k moves outputs; (x-h) moves inputs.

Linear example

Let f(x)=2x-3.

Vertical shift up 5:

g(x)=f(x)+5 = (2x-3)+5 = 2x+2

Horizontal shift right 4:

h(x)=f(x-4)=2(x-4)-3=2x-8-3=2x-11

Both (right 4, up 5):

p(x)=f(x-4)+5 = 2(x-4)-3+5 = 2x-6

Key-point mapping idea: If you know two points on the line (for example, intercepts), shift those points and draw the new line through them.

Quadratic example (vertex is a key point)

Let f(x)=x^2. Useful key points include (-2,4), (-1,1), (0,0), (1,1), (2,4).

Define g(x)=f(x-3)+2=(x-3)^2+2.

• Right 3, up 2.
• Vertex (0,0) maps to (3,2).
• (1,1) maps to (4,3); (-1,1) maps to (2,3).

Plot the mapped points and sketch the same parabola shape opening upward.

Absolute value example (corner point is key)

Let f(x)=|x|. Useful key points: (-2,2), (-1,1), (0,0), (1,1), (2,2).

Define g(x)=f(x+2)-3=|x+2|-3.

• x+2 means left 2.
• -3 means down 3.
• Corner (0,0) maps to (-2,-3).
• (2,2) maps to (0,-1); (-2,2) maps to (-4,-1).

Plot the mapped corner and points, then draw the same V-shape.

Exercises: Write the Transformed Equation and Sketch Using Mapped Key Points

A. Write the transformed function

In each problem, start with the given base function f(x). Write the new function g(x) after the described shift. Use function notation (do not expand unless asked).

1. Base: f(x). Shift the graph up 7 units. Write g(x)=

2. Base: f(x). Shift the graph down 4 units. Write g(x)=

3. Base: f(x). Shift the graph right 5 units. Write g(x)=

4. Base: f(x). Shift the graph left 3 units. Write g(x)=

5. Base: f(x). Shift right 2 and up 6. Write g(x)=

6. Base: f(x). Shift left 4 and down 1. Write g(x)=

B. Use point mapping to sketch (do not re-plot from scratch)

For each problem, the listed points lie on y=f(x). Map them to points on the transformed graph and sketch by connecting with the same shape.

1. Transformation: g(x)=f(x)+3. Original points: (-2,-1), (0,2), (3,0). List the new points.

2. Transformation: g(x)=f(x-4). Original points: (-1,5), (2,1), (4,-2). List the new points.

3. Transformation: g(x)=f(x+2)-6. Original points: (0,0), (1,3), (-2,4). List the new points.

4. Transformation: g(x)=f(x-1)+2. Original points: (-3,2), (0,-1), (2,4). List the new points.

C. Apply shifts to specific families (consistent method)

1. Linear: Start with f(x)=3x+1. Write the equation of the graph shifted left 2 and down 5.

2. Quadratic: Start with f(x)=x^2. Write the equation of the graph shifted right 4 and up 1. Then list the transformed locations of the key points (-1,1), (0,0), (1,1).

3. Absolute value: Start with f(x)=|x|. Write the equation of the graph shifted left 3 and up 2. Then list the transformed locations of (0,0), (2,2), and (-2,2).

4. Mixed check (interpretation): A graph of y=f(x) is moved down 2 and right 7 to create y=g(x). Write g(x) in terms of f.