Question Statement

Draw the graph of the following modulus functions:

(i) $f(x)=-1.5|x|$

(ii) $f(x)=1+2|x|$

(iii) $f(x)=3|x|+x$

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Background and Explanation

The modulus function, denoted by $|x|$, represents the absolute value of $x$, meaning $|x|=x$ if $x\ge 0$ and $|x|=-x$ if $x<0$. To graph these functions, we calculate a set of coordinates by substituting various values of $x$ into the function and then plotting these points on a Cartesian plane to observe the resulting V-shaped or piecewise linear paths.

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Solution

Part (i) $f(x)=-1.5|x|$

To graph this function, we evaluate $f(x)$ for several values of $x$. Since the coefficient of the modulus is negative ($-1.5$), we expect the graph to be an inverted "V" shape opening downwards.

Table of Values:

Reasoning:

• For $x=-2$, $f(-2)=-1.5|-2|=-1.5(2)=-3$.
• For $x=0$, $f(0)=-1.5|0|=0$.
• For $x=2$, $f(2)=-1.5|2|=-3$.

The graph passes through the origin $(0,0)$ and decreases linearly as $x$ moves away from zero in either direction.

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Part (ii) $f(x)=1+2|x|$

In this function, the modulus is multiplied by $2$ and then $1$ is added. This results in a steeper "V" shape that is shifted upwards by $1$ unit on the y-axis.

Table of Values:

Reasoning:

• For $x=-2$, $f(-2)=1+2|-2|=1+2(2)=5$.
• For $x=0$, $f(0)=1+2|0|=1$. This is the vertex of the graph.
• For $x=2$, $f(2)=1+2|2|=5$.

The graph is symmetric about the y-axis with its lowest point at $(0,1)$.

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Part (iii) $f(x)=3|x|+x$

This function combines a modulus term with a linear term. This creates an asymmetrical graph because the slope changes depending on whether $x$ is positive or negative.

Table of Values:

Reasoning:

• If $x\ge 0$, $f(x)=3(x)+x=4x$. (A steep line with slope 4).
• If $x<0$, $f(x)=3(-x)+x=-2x$. (A line with slope -2).
• For $x=-3$, $f(-3)=3|-3|+(-3)=9-3=6$.
• For $x=3$, $f(3)=3|3|+3=9+3=12$.

The graph has a vertex at $(0,0)$ but rises more sharply on the right side than it does on the left.

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Key Formulas or Methods Used

• Definition of Modulus: $|x|=x$ if $x\ge 0$ and $|x|=-x$ if $x<0$.
• Point-Plotting Method: Choosing a set of $x$-values, calculating $y$-values, and plotting them to visualize the function.
• Piecewise Analysis: Breaking the function into two linear parts based on the sign of the expression inside the modulus.

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Summary of Steps

1. Select $x$-values: Choose a range of values including negative numbers, zero, and positive numbers to see the full shape of the modulus graph.
2. Calculate $f(x)$: Substitute each $x$ into the given equation, remembering that $|x|$ always returns a non-negative value.
3. Create a Table: Organize the $(x,f(x))$ pairs into a table for easy reference.
4. Plot and Connect: Plot the points on a coordinate plane and connect them with straight lines, noting the "corner" or vertex where the expression inside the modulus equals zero.