What shape is the graph of y=|ax+b|?

A V-shape with its vertex on the x-axis.

Where is the vertex of y=|ax+b|?

At \left(-\frac ba,0\right), where ax+b=0.

How do you sketch y=|f(x)| from y=f(x)?

Reflect any part below the x-axis in the x-axis.

Write y=|3-x| without modulus signs.

y=3-x for x\le3 and y=x-3 for x3.

Why must solutions be checked when solving branch by branch?

Each solution must lie in the interval where that branch applies.

The modulus functionAQA A-Level Maths: Flashcards

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Define $|x|$ for $x\ge0$ and $x<0$.

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Define $|x|$ for $x\ge0$ and $x<0$ .: $|x|=x$ if $x\ge0$ and $|x|=-x$ if $x<0$ .
What is the range of $y=|ax+b|$ ?: $y\ge0$ .
What shape is the graph of $y=|ax+b|$ ?: A V-shape with its vertex on the $x$ -axis.
Where is the vertex of $y=|ax+b|$ ?: At $\left(-\frac ba,0\right)$ , where $ax+b=0$ .
What is the $y$ -intercept of $y=|ax+b|$ ?: $|b|$ .
How do you sketch $y=|f(x)|$ from $y=f(x)$ ?: Reflect any part below the $x$ -axis in the $x$ -axis.
What are the gradients of the two branches of $y=|ax+b|$ ?: $a$ and $-a$ .
Equation of the line of symmetry of $y=|2x-6|$ ?: $x=3$ .
Vertex of $y=|2x+8|$ ?: $(-4,0)$ .
How many solutions does $|ax+b|=k$ have for $k>0$ , $k=0$ , $k<0$ ?: Two, one and none.
Write $y=|3-x|$ without modulus signs.: $y=3-x$ for $x\le3$ and $y=x-3$ for $x>3$ .
Why must solutions be checked when solving branch by branch?: Each solution must lie in the interval where that branch applies.

Exam questions on The modulus function

The graph of $y=|2x-6|$ is considered.
Write $y=|2x-6|$ without modulus signs, stating the values of $x$ for which each form applies.2 marks
The curve $y=|x+4|$ and the horizontal line $y=k$ , where $k$ is a constant, are considered.
Find the coordinates of the points where the curve $y=|x+4|$ meets the axes.2 marks
The curve $C$ has equation $y=|4-3x|$ .
Find the coordinates of the vertex of $C$ and of the point where $C$ meets the $y$ -axis, and state the equation of the line of symmetry of $C$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).