Linear Transformations: shifts, reflections, dilations

Shifting and Reflecting Linear Functions

Explore the transformation of linear functions through shifting, reflecting, and dilating. Learn how to manipulate equations to create new graphs and understand the impact of these transformations on the function's behavior.

Introduction to Linear Functions

• Linear functions are mathematical functions that can be represented by a straight line.

• They have the form y = mx + b, where m is the slope and b is the y-intercept. Also written as f(x) = mx + b.

• Shifting a linear function involves changing the value of b.

• Reflecting a linear function involves changing the sign of m.

• Dilating a linear function involves multiplying the function by a number other than 0 and 1.

Multiple Choice

What are the two ways to transform a linear function?

• Changing the values of m and b

• Changing the sign of b

• Changing the values of x and y

• Changing the sign of x or y

Transforming Linear Functions

Trivia: Did you know that there are two ways to transform a linear function? One way is by changing the values of m and/or b. Another way is by changing the sign of m. These transformations can alter the slope and y-intercept of the function, resulting in a different graph.

Understanding Linear Function Shifts

• Vertical shifts are determined by adding or subtracting a value from the variable.

• Using the formula g(x) = a*f(x) + k, the variable k represents the vertiKal shift of the original function.

• Example: g(x) = f(x) - 5

• This equation represents the function f being shifted down 5 units to create function g.

Multiple Choice

What determines the direction of a vertical shift in a linear function?

• Adding or subtracting a value from the variable

• Multiplying or dividing a value from the variable

• Positive shifts

• Negative shifts

Vertical Shifts:

Trivia: The direction of a vertical shift in a linear function is determined by adding or subtracting a value from the variable. Adding shifts the function up and subtracting shifts the function down.

Vertical Shifts

• Vertical shifts involve moving a linear function up or down along the y-axis.

• The amount of shift is determined by the value added or subtracted from the function.

• Vertical shifts do not affect the slope of the function.

• A vertical shift creates a parallel function to the original.

Multiple Choice

What is the effect of vertical shifts on the slope of a linear function?

• Vertical shifts increase the slope of the function until the lines are perpendicular.

• Vertical shifts decrease the slope of the function.

• Vertical shifts have no effect on the slope of the function, making the lines parallel.

• Vertical shifts make the slope undefined.

Vertical Shifts

Trivia: Did you know that vertical shifts have no effect on the slope of a linear function? The slope remains the same regardless of any vertical shifts. This means that the steepness of the line remains constant, only its position on the y-axis changes. So, vertical shifts only affect the y-intercept, not the slope!

Reflections of Linear Functions

• Reflections of linear functions involve flipping the graph over an axis.

• Reflections preserve the slope of the original function but change the sign of the y-intercept.

• Use the formula g(x) = -f(x) to reflect a linear function f(x) over the x-axis.

Multiple Choice

What formula can be used to reflect a linear function over the x-axis?

• p(x) = -f(x)

• y = f(x)

• k(x) = f-(x)

• y = f(-x)

Reflecting Linear Functions

Trivia: Did you know that to reflect a linear function over the x-axis, you can use the formula y = -f(x)?

Reflecting Linear Functions

• Reflecting over the x-axis is a transformation that flips a linear function upside down.

• To reflect a linear function over the x-axis, multiply the function by -1.

• For example, reflecting the function y = 2x + 3 over the x-axis results in y = -2x - 3.

Multiple Choice

What happens to the y-intercept and slope when a linear function is reflected over the x-axis?

• They remain the same

• They change signs

• They become positive

• They become zero

Reflection over x-axis

Trivia: When a linear function is reflected over the x-axis, the y-intercept and slope change signs. It is like flipping the function upside down!

Dilating Linear Functions

Multiple Choice

Which function represents the steepest line if f(x) = x?

• j(x) = f(x) + 10

• h(x) = 5*f(x)

• w(x) = -7*f(x)

• d(x) = 2/3*f(x)

Dilating Linear Functions

Trivia: When a linear function is multiplied, by a (positive or negative) number greater than 1, it increases the steepness!



-7x is steeper than 5x, it is just slanted in a different direction.