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How to reflect a graph through the x-axis

Precalculus: Master Advanced Math Concepts

Functions

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One to one functions

Identifying functions

Introduction to linear equations

Function notation

Composite functions

Adding functions

Subtracting functions

Multiplying functions

Dividing functions

Operations of Functions – In a Nutshell

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

Transformations of functions: Horizontal stretches

Transformations of functions: Vertical stretches

Even and odd functions

Difference quotient: applications of functions

Polynomials

Descartes&apos; rule of signs

Descartes' Rule of Signs

Descartes' rule of signs

Descartes" rule of signs

Polynomial long division

Polynomial synthetic division

Pascal's triangle

Pascal's Triangle

Pascal"s Triangle

Pascal"s triangle

Binomial theorem

What is a polynomial function?

Polynomial functions

Recognizing a Polynomial Function

Factor theorem

Using the Factor Theorem to Test For Factors of a Polynomial

Rational zero theorem

Rational Zero Theorem

Characteristics of polynomial graphs

Determining the equation of a polynomial function

Applications of polynomial functions

Multiplicities of polynomials

Imaginary zeros of polynomials

Solving polynomial inequalities

Rational Functions

What is a rational function?

Point of discontinuity

Slant asymptote

Determining the Equation of a Slant Asymptote Using Long Division

Determining the Equation of a Slant Asymptote Using Synthetic Division

Solving rational inequalities

Solving Rational Inequalities With Two Fractions

Vertical asymptote

Horizontal asymptote

Horizontal Asymptote

Graphs of rational functions

Radical

Transformations of radical functions

Square root of a function

Solving radical equations

Solving Radical Equations Involving One Square Root Term

Solving Radical Equations Involving More than One Square Root Term

Basic radical functions

Basic Radical Function

Basic Radical Functions

Roots and Rational Exponents

Evaluating and simplifying radicals

Power of a power rule

Simplify the power

Combining the exponent rules

Rational exponents

Convert between radicals and rational exponents

Square and square roots

Square Root of a Perfect Square

Converting radicals to mixed radicals

Converting radicals to entire radicals

Systems of Equations

Solving 3 variable systems of equations by substitution

Solving 3 variable systems of equations by elimination

Word problems relating 3 variable systems of equations

Solving 3 variable systems of equations with no or infinite solutions

Solving 3 variable systems of equations (no solution, infinite solutions)

System of Equations With No Solution

System of Equations With Infinite Solutions

System of Equations With Infinite Solutions - Extended

Determining number of solutions to linear equations

Solving linear systems by graphing

Solving systems of linear equations by graphing

Using elimination method to solve systems of equations

Solving systems of linear equations by elimination

Using substitution method to solve systems of equations

Solving systems of linear equations by substitution

System of Inequalities

What is linear programming?

Finding Optimal Values Given Constraints In Slope-Intercept Form

Finding Optimal Values Given Constraints In Standard Form

Linear programming word problems

Shortcut: The vertex formula

Shortcut: Vertex formula

Converting general form into vertex form by applying the vertex formula

Vertex Formula Application

Solving absolute value inequalities

Solving Absolute Value Inequalities Involving "less than"

Solving Absolute Value Inequalities Involving "greater than"

Recognizing Absolute Value Inequalities with "No Solution" or "All Real Numbers"

Graphing linear inequalities in two variables

Graphing quadratic inequalities in two variables 

Unit circle

Quotient identities and reciprocal identities

Pythagorean identities

Sum and difference identities

Sum and Difference Identities

Double-angle identities

Use sine ratio to calculate angles and side (Sin = \( \frac{o}{h}\) )

Use cosine ratio to calculate angles and side (Cos = \( \frac{a}{h}\) )

Combination of SohCahToa questions

Word problems relating ladder in trigonometry

Word problems relating guy wire in trigonometry

Use tangent ratio to calculate angles and side (Tan = \( \frac{o}{a}\) )

Solving first degree trigonometric equations

Law of sines

Ambiguous case: SSA triangles

Law of cosines

Angle in standard position

Coterminal angles

Coterminal Angles

Reference angle

Find the exact value of trigonometric ratios

ASTC rule in trigonometry (All Students Take Calculus)

ASTC rule in trigonometry

ASTC rule in trigonometry (All Students Take Calculus)

Converting between degrees and radians

Trigonometric ratios for angles in radians

Trigonometric ratios of angles in radians

Radian measure and arc length

Sine graph: y = sin x

Cosine graph: y = cos x

Tangent graph: y = tan x

Cotangent graph: y = cot x

Secant graph: y = sec x

Cosecant graph: y = csc x

Graphing transformations of trigonometric functions

Determining trigonometric functions given their graphs

Determining non-permissible values for trig expressions

Sequences and Series

Arithmetic sequences

Arithmetic series

Arithmetic series formula

Geometric sequences

Geometric Series

Geometric series

Infinite geometric series

Sigma notation

Influencing factors in data collection

Median and mode

Median and Mode Table

Median and Mode – Word Problem

Median of an even length list

Median of an odd length list

Mean

Mean Word Problems

Find the Mean

Application of Means

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https://dmn92m25mtw4z.cloudfront.net/img/textbook/textbook-precalculus.png

Reflection across the x-axis: y = -f(x)

What You'll Learn

Recognize that y = -f(x) creates a reflection across the x-axis

Understand that the negative sign outside f(x) belongs to y, not x

Apply the rule that all y-coordinates divide by -1 during x-axis reflection

Identify invariant points that remain unchanged after reflection

Distinguish between x-axis reflections (y = -f(x)) and y-axis reflections (y = f(-x))

What You'll Practice

Reflecting graphs across the x-axis by transforming coordinate points

Converting positive y-values to negative y-values and vice versa

Plotting reflected points and sketching transformed functions

Verifying reflections by comparing original and reflected coordinates

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

2 Video lessons

Learning resources

Skills

Reflections

Function Transformations

Coordinate Geometry

Graphing

Invariant Points

Negative Functions