Graphs and Modulus Intersections | CAIE 9709 Pure Mathematics 2 Notes

Pure Mathematics 2

Mathematics9709

Reflect only the part below the x-axis

To draw $y=|2x-4|$ , first draw $y=2x-4$ . Positive outputs stay unchanged; negative outputs become their opposites. The $x$ -coordinates do not move.

The reflected point keeps its x-coordinate and changes the sign of its y-coordinate.

The vertex is the point where the two straight branches meet. Set the inside to zero: $2x-4=0$ , giving vertex $(2,0)$ . Set $x=0$ for the y-intercept: $(0,4)$ .

The sign changes at 2. To its left, negate the entire expression; to its right, leave it unchanged:

|2x-4|=\begin{cases}4-2x&x<2,\\2x-4&x\ge2.\end{cases}

The branches have gradients −2 and 2. For $|ax+b|$ with $a\ne0$ , the vertex is $(-b/a,0)$ and the y-intercept is $(0,|b|)$ . If $a=0$ , the graph is the horizontal line $y=|b|$ , not a V.

Your turnPractice

Sketch $y=|3-x|$ and write its two branch rules. Label both intercepts.

Show worked answer

|3-x|=\begin{cases}3-x&x\le3,\\x-3&x>3.\end{cases}

A V with vertex $(3,0)$ , y-intercept $(0,3)$ , gradients −1 and 1. Both rules give zero at 3, so either branch may include that endpoint.

Intersections solve equations

At an intersection, two graphs have the same $x$ and the same $y$ . Their x-coordinates therefore solve the equation formed by equating the two outputs.

Worked example

Where does y = |2x − 4| meet y = 2?

\begin{gathered}2x-4=2\\\text{or}\\2x-4=-2\end{gathered}

The roots are 3 and 1; the intersection points are $(3,2)$ and $(1,2)$ . A sketch checks the count; algebra supplies exact coordinates.

Your turnPractice

Find the intersection of $y=|x-4|$ and $y=2x+1$ . Why is $x=-5$ not a second intersection?

Show worked answer

Answer

(1,3)

At −5 the graphs have y-values 9 and −9, which are unequal. The extension of one straight branch is not the modulus graph on that side.

Reflect only the part below the x-axis

To draw $y=|2x-4|$ , first draw $y=2x-4$ . Positive outputs stay unchanged; negative outputs become their opposites. The $x$ -coordinates do not move.

The reflected point keeps its x-coordinate and changes the sign of its y-coordinate.

The vertex is the point where the two straight branches meet. Set the inside to zero: $2x-4=0$ , giving vertex $(2,0)$ . Set $x=0$ for the y-intercept: $(0,4)$ .

The sign changes at 2. To its left, negate the entire expression; to its right, leave it unchanged:

|2x-4|=\begin{cases}4-2x&x<2,\\2x-4&x\ge2.\end{cases}

The branches have gradients −2 and 2. For $|ax+b|$ with $a\ne0$ , the vertex is $(-b/a,0)$ and the y-intercept is $(0,|b|)$ . If $a=0$ , the graph is the horizontal line $y=|b|$ , not a V.

Your turnPractice

Sketch $y=|3-x|$ and write its two branch rules. Label both intercepts.

Show worked answer

|3-x|=\begin{cases}3-x&x\le3,\\x-3&x>3.\end{cases}

A V with vertex $(3,0)$ , y-intercept $(0,3)$ , gradients −1 and 1. Both rules give zero at 3, so either branch may include that endpoint.

Intersections solve equations

At an intersection, two graphs have the same $x$ and the same $y$ . Their x-coordinates therefore solve the equation formed by equating the two outputs.

Worked example

Where does y = |2x − 4| meet y = 2?

\begin{gathered}2x-4=2\\\text{or}\\2x-4=-2\end{gathered}

The roots are 3 and 1; the intersection points are $(3,2)$ and $(1,2)$ . A sketch checks the count; algebra supplies exact coordinates.

Your turnPractice

Find the intersection of $y=|x-4|$ and $y=2x+1$ . Why is $x=-5$ not a second intersection?

Show worked answer

Answer

(1,3)

At −5 the graphs have y-values 9 and −9, which are unequal. The extension of one straight branch is not the modulus graph on that side.