Inverses and Graphing Linear Functions
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Questions and Answers

The preimage of point A is located at (4, ______)

2

The transformation for point B results in the image at B' with coordinates (______, 5).

-6

The vector that represents the transformation is (______, 2) for the y-coordinate.

−4

After the transformation, point D maps to D' at coordinates (______, 1).

<p>-3</p> Signup and view all the answers

The coordinates of point C' after the mapping are (______, 2).

<p>-6</p> Signup and view all the answers

The preimage of point B has coordinates (______, 3).

<p>-2</p> Signup and view all the answers

The transformation rule involves subtracting 4 from the x-coordinate and adding ______ to the y-coordinate.

<p>2</p> Signup and view all the answers

Point A' is located at the coordinates (______, 4).

<p>0</p> Signup and view all the answers

The transformation formula used is (x, y) → (x + 2, y - ______)

<p>3</p> Signup and view all the answers

The image of point A after the transformation is A'(______, 1)

<p>3</p> Signup and view all the answers

Point B has coordinates (______, -3)

<p>4</p> Signup and view all the answers

After transformation, the coordinates for B' are (6, ______)

<p>-6</p> Signup and view all the answers

Point C is initially located at (______, -2)

<p>-3</p> Signup and view all the answers

The transformed image of point C is C'([-1], ______)

<p>-5</p> Signup and view all the answers

Point D starts at (-1, ______)

<p>3</p> Signup and view all the answers

The translation vector used in the example is ______, ______.

<p>-2, 5</p> Signup and view all the answers

The image of point A(4, -2) after translation becomes point A' ______, ______.

<p>2, 3</p> Signup and view all the answers

Point B is translated from B(1, -1) to B' ______, ______.

<p>-1, 4</p> Signup and view all the answers

C(-3, 2) transforms into C' ______, ______ after translation.

<p>-5, 7</p> Signup and view all the answers

The preimage of point A is A(4, -2) and its image after translation is A' ______, ______.

<p>2, 3</p> Signup and view all the answers

In vector mapping, the original coordinates (x, y) transform according to the rule (x, y) → ______, ______.

<p>(x - 2, y + 5)</p> Signup and view all the answers

Using the translation vector <3, -1>, the new coordinates for point A(-2, 4) will be ______, ______.

<p>(1, 3)</p> Signup and view all the answers

The geometric concept being used in the question is known as ______ mapping.

<p>vector</p> Signup and view all the answers

If a point is at (x, y) and is translated by vector (-2, 5), its new coordinates are ______, ______.

<p>(x - 2, y + 5)</p> Signup and view all the answers

In the context of geometry, the process of moving points or figures is called ______.

<p>translation</p> Signup and view all the answers

Flashcards

Image

The resulting point or figure after a transformation

Preimage

The original point or figure before a transformation

Transformation

A change in the position, size, or shape of a figure.

Coordinate Rule

A set of instructions that describe how to move points to get the image of a figure.

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Translation

A transformation that slides all points of a figure the same distance in the same direction.

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Component Form

A way of representing a vector that maps a point to another point by specifying the horizontal and vertical distances.

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Vector

A quantity that has both magnitude (size) and direction.

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Mapping

A way to show corresponding parts between the preimage and the image after a transformation.

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Transformation rule (x,y) → (x+2, y-3)

A set of instructions that shifts each point on a coordinate plane to a new location.

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Pre-image point

The original point before a transformation.

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Image point

The new point after a transformation.

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Coordinate change (x, y)

Shifting a point horizontally (x) and vertically (y).

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x-coordinate

The horizontal position of a point on a graph. Its value.

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y-coordinate

The vertical position of a point on a graph. Its value.

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Translation of a point

Moving a point a fixed distance and direction in a coordinate plane.

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Image of a point

The new location of a point after a transformation (like translation).

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Coordinate Plane

A two-dimensional plane where points are located using x and y coordinates.

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Translation rule

A rule describing how to move a point to its image.

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Translation vector <2, 5>

Moves points 2 units horizontally and 5 units vertically.

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Study Notes

Inverses of Linear Functions

  • To find the inverse of a linear function, switch the x and y in the equation.
  • First, rewrite the function using a y instead of a function notation.
  • Switch the x and the y, and solve for y.

Graphing Inverses

  • The inverse of a function is the reflection of the function over the line y = x.
  • Graph the inverse of f(x) on the same axes as f(x).

Determining If Two Functions Are Inverses

  • Use functional composition to determine if two functions are inverses.
  • Show that the function f(x) = 2x - 5 is the inverse of g(x) = (x + 2)/2
  • Show that the functions f(x) = x + 6 and g(x) = x - 6 are inverses.
  • Show that f(x) = -x + 7 and g(x) = -x - 7 are not inverses.

Graphing Absolute Value Functions

  • Understand translations in absolute value functions
  • The vertex of f(x) = |x| is (0, 0).
  • Given the graph of f(x) = |x|, you can use what you know about horizontal and vertical translations to graph g(x) = |x-h|+k
  • The vertex of g(x) is (h,k)
  • The slope of the left branch and right branch of f(x) and g(x) is 1

Graphing Transformations of Absolute Value Functions

  • The sign of 'a' determines if there is a vertical stretch or vertical compression/reflection
  • The slope of g(x) is a for values of x > h and -a for x< h
  • The vertex of g(x) = a|x-h| + k is (h, k)
  • The slope of the right branch of the absolute value function is 'a'. The slope of the left branch is '-a'

Graphing Linear Inequalities

  • Determine the equation for the boundary line.
  • Determine the inequality symbol.
  • Shade the region that satisfies the inequality.
  • Check your results to ensure that a point in the solution region makes the inequality true.
  • Check that a point on the boundary line makes the inequality false.
  • Draw a dashed line for < or >. Draw a solid line for ≤ or ≥

Linear Inequalities in Two Variables

  • Write a linear inequality to describe a given situation.
  • Graph the boundary line of the inequality.
  • Shade the region that represents the solution set of the inequality.
  • Interpret or check the solutions to the inequality.

Vectors and Transformations

  • Identify the vector that maps one polygon onto another.
  • This includes identifying the preimage and the image.
  • Calculate the differences between the vertices/points of the preimage and image.
  • Use a table to find the differences.
  • Draw the figure with the given vertices.
  • Draw the image after a translation by the given vector.

Rotations

  • Find the center of rotation and angle of rotation

  • Draw segments from the center of rotation to corresponding vertices in the preimage and its image.

  • Measure the angle between the segments to determine the angle of rotation.

  • Describe any rotations less than 360° that map the polygon onto itself.

  • Determine the degree of rotation

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Related Documents

Inverse of Linear Functions PDF

Description

This quiz covers the concept of inverses in linear functions, including how to find and graph them. You'll also explore how to determine if two functions are inverses through functional composition. Finally, the quiz touches on graphing absolute value functions and their translations.

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