MYP Extended Mathematics Topic Reflections Revision Notes

MYP Extended Mathematics Topic Reflections Revision Notes | RevisionDojo

Reflections Are Mirror Transformations Of Graphs

A reflection is often the quickest way to generate a new graph because you do not need to recalculate lots of points, you "mirror" what you already know.
In coordinate geometry, the "mirror line" is an axis.
Reflecting in an axis changes the sign of one coordinate while keeping the other the same.

Analogy

Think of the x-axis as the surface of a perfectly still lake.
The graph above the lake has an identical "upside-down" image below the surface.
That is what reflecting in the x-axis does.

Reflecting In The X-Axis Changes The Sign Of All y-Values

When you reflect a point $(x,y)$ in the x-axis, its horizontal position does not change, but its vertical position flips. $$ (x,y) \mapsto (x,-y) $$
So, for a whole function $y=f(x)$, the reflected graph has equation $$ y=-f(x) $$
This matches an important generalization:

The graph of $y=-f(x)$ is a reflection of the graph of $y=f(x)$ in the x-axis.

How To Sketch $y=-f(x)$ From A Known Graph

Keep each $x$-coordinate the same.
Multiply each $y$-value by $-1$.
Plot the mirrored points and join with the same shape.

Example

Suppose $f(x)=3x+2$.
Reflecting in the x-axis gives $$y=-f(x)=-(3x+2)=-3x-2$$
Check with intercepts: the original line crosses the y-axis at $(0,2)$, the reflection crosses at $(0,-2)$.

Common Mistake

A reflection in the x-axis is not a left-right flip.
Students often confuse $-f(x)$ with $f(-x)$.
The negative sign outside the function changes y-values.

Example

If $f(x)=(x-2)^2+3$, then $$-f(x)=-(x-2)^2-3.$$
Notice what happens to key features:
The vertex $(2,3)$ becomes $(2,-3)$.
A parabola that opened upward now opens downward.

Reflecting In The Y-Axis Changes The Sign Of All x-Values

When you reflect a point $(x,y)$ in the y-axis, its vertical position does not change, but its horizontal position flips. $$ (x,y) \mapsto (-x,y) $$
So, for a whole function $y=f(x)$, the reflected graph has equation $$ y=f(-x) $$
This matches another key generalization:

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Reflections Are Mirror Transformations Of Graphs

A reflection is often the quickest way to generate a new graph because you do not need to recalculate lots of points, you "mirror" what you already know.
In coordinate geometry, the "mirror line" is an axis.
Reflecting in an axis changes the sign of one coordinate while keeping the other the same.

Analogy

Think of the x-axis as the surface of a perfectly still lake.
The graph above the lake has an identical "upside-down" image below the surface.
That is what reflecting in the x-axis does.

Reflecting In The X-Axis Changes The Sign Of All y-Values

When you reflect a point $(x,y)$ in the x-axis, its horizontal position does not change, but its vertical position flips. $$ (x,y) \mapsto (x,-y) $$
So, for a whole function $y=f(x)$, the reflected graph has equation $$ y=-f(x) $$
This matches an important generalization:

The graph of $y=-f(x)$ is a reflection of the graph of $y=f(x)$ in the x-axis.

How To Sketch $y=-f(x)$ From A Known Graph

Keep each $x$-coordinate the same.
Multiply each $y$-value by $-1$.
Plot the mirrored points and join with the same shape.

Example

Suppose $f(x)=3x+2$.
Reflecting in the x-axis gives $$y=-f(x)=-(3x+2)=-3x-2$$
Check with intercepts: the original line crosses the y-axis at $(0,2)$, the reflection crosses at $(0,-2)$.

Common Mistake

A reflection in the x-axis is not a left-right flip.
Students often confuse $-f(x)$ with $f(-x)$.
The negative sign outside the function changes y-values.

Example

If $f(x)=(x-2)^2+3$, then $$-f(x)=-(x-2)^2-3.$$
Notice what happens to key features:
The vertex $(2,3)$ becomes $(2,-3)$.
A parabola that opened upward now opens downward.

Reflecting In The Y-Axis Changes The Sign Of All x-Values

When you reflect a point $(x,y)$ in the y-axis, its vertical position does not change, but its horizontal position flips. $$ (x,y) \mapsto (-x,y) $$
So, for a whole function $y=f(x)$, the reflected graph has equation $$ y=f(-x) $$
This matches another key generalization:

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

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Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Reflections Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Reflections Are Mirror Transformations Of Graphs

Reflecting In The X-Axis Changes The Sign Of All y-Values

How To Sketch $y=-f(x)$ From A Known Graph

Reflecting In The Y-Axis Changes The Sign Of All x-Values

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Reflections Are Mirror Transformations Of Graphs

Reflecting In The X-Axis Changes The Sign Of All y-Values

How To Sketch $y=-f(x)$ From A Known Graph

Reflecting In The Y-Axis Changes The Sign Of All x-Values

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account