How to Transform Linear Functions
Transforming linear functions refers to the process of changing the shape or position of a linear function, while still preserving its linearity. This can be done by applying certain operations, such as translation, reflection, dilation, and rotation, to the function.
Tutor-style math help
Transform Linear Functions: what to notice and how to work it
Functions skill
A function is a rule that gives each input exactly one output. Function notation, tables, graphs, and equations are different ways to show the same input-output relationship.
What to notice first
Ask what kind of input you are given. Sometimes you substitute a number, sometimes you read a graph, and sometimes you combine two rules.
Common student mistake
Do not read \(f(4)\) as multiplication. It means the output of f when the input is 4.
Key formulas and cues
\(f(a)\text{ means replace }x\text{ with }a\)
\((f\circ g)(x)=f(g(x))\)
\(f^{-1}(x)\text{ reverses }f(x)\)
A reliable path
- Identify the inputFind the x-value, expression, or inner function being used.
- Apply the ruleSubstitute with parentheses so signs and powers stay clear.
- Interpret the outputState the value, point, interval, domain, range, or inverse relationship.
Worked examples
Evaluate a function
Example: \(f(x)=4x-3\), find \(f(2)\)
- Replace x with 2.
- Compute 4(2) – 3.
- Simplify.
Answer: \(5\)
Compose functions
Example: \(f(x)=x+1\), \(g(x)=2x\), find \(f(g(3))\)
- Find g(3) = 6.
- Use that as the input for f.
- f(6) = 7.
Answer: \(7\)
Try one before moving on
Try: If \(h(x)=2x^2\), find \(h(-3)\).
Answer: \(18\). Use parentheses: \(2(-3)^2=18\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Transform Linear Functions: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. \(f(3)\) means:
2. A function gives each input:
3. If \(f(x)=x+4\), then \(f(6)=\)
Related Topics
- How to Transform Quadratic Equations
- Transformation on the Coordinate Plane: Dilation
- Transformation on the Coordinate Plane: Rotation
- Transformation on the Coordinate Plane: Reflection
- Transformation: Rotations, Reflections, and Translations
Step-by-step to Transform Linear Functions
Here are some examples of how to transform linear functions:
- Translation: To translate a linear function up or down, you can add or subtract a constant from the \(y\)-coordinate. For example, \(y = 2x + 1\) can be translated up \(2\) units by adding \(2\) to the \(y\)-coordinate, resulting in \(y = 2x + 3\).
- Reflection: To reflect a linear function across the \(x\)-axis, you can negate the \(y\)-coordinate. For example, \(y = 2x + 1\) can be reflected across the \(x\)-axis by negating the \(y\)-coordinate, resulting in \(y = -2x – 1\).
- Dilation: To dilate a linear function, you can multiply the \(x\)-coordinate by a constant factor. For example, \(y = 2x + 1\) can be dilated by a factor of \(2\) by multiplying the \(x\)-coordinate by \(2\), resulting in \(y = 4x + 1\).
- Rotation: To rotate a linear function, you can change the slope of the function. For example, \(y = 2x + 1\) can be rotated \(90\) degrees by making the slope \(-\frac{1}{2}\), resulting in \(y = -\frac{1}{2}x + 1\)
Note: It’s important to notice that transformations does not change the type of function, it only changes the shape and position of the function.
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