Question Statement

Plot the graph of the following functions:

(i) $f(x)=3x-2$ (ii) $f(x)=3x$ (iii) $f(x)=1-2x$ (iv) $g(x)=x^2+4$ (v) $g(x)=x^2-x-6$ (vi) $g(x)=\sqrt{2x+1}$

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Background and Explanation

To plot the graph of a function, we determine several $(x,f(x))$ coordinate pairs by substituting chosen $x$-values into the function. For linear functions, a few points are sufficient to draw a straight line, while for quadratic and square root functions, more points are needed to accurately capture the curve and key features like vertices or endpoints.

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Solution

Part (i) $f(x)=3x-2$

To plot this linear function, we calculate the values of $f(x)$ for various $x$ inputs:

Calculations: $\begin{array}{rl}f(-3)& =3(-3)-2\\ & =-9-2\\ & =-11\end{array}$ $\begin{array}{rl}f(-2)& =3(-2)-2\\ & =-6-2\\ & =-8\end{array}$ $\begin{array}{rl}f(0)& =3(0)-2\\ & =-2\end{array}$ $\begin{array}{rl}f(2)& =3(2)-2\\ & =6-2=4\end{array}$

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Part (ii) $f(x)=3x$

This is a linear function passing through the origin $(0,0)$. We can easily draw the table of values:

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Part (iii) $f(x)=1-2x$

We substitute $x$ values into the linear equation to find the corresponding $y$ coordinates:

Calculations: $\begin{array}{rl}f(-2)& =1-2(-2)=1+4=5\\ f(-1)& =1-2(-1)=1+2=3\\ f(0)& =1-2(0)=1\\ f(1)& =1-2(1)=-1\\ f(2)& =1-2(2)=-3\end{array}$

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Part (iv) $g(x)=x^2+4$

In order to sketch the graph of a quadratic function, we should identify the vertex (minimum or maximum point). For $g(x)=x^2+4$, the minimum point is at $(0,4)$. We calculate additional points to define the parabola:

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Part (v) $g(x)=x^2-x-6$

For this quadratic, we include the value $x=1/2$ to find the vertex of the parabola.

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Part (vi) $g(x)=\sqrt{2x+1}$

For a square root function, the value inside the root must be non-negative ($2x+1\ge 0⟹x\ge -1/2$). We choose $x$ values that result in easy-to-plot square roots where possible:

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Key Formulas or Methods Used

• Substitution: Replacing the variable $x$ with a specific value to find $f(x)$.
• Linear Functions: Graphs of the form $f(x)=mx+c$ result in straight lines.
• Quadratic Functions: Graphs of the form $g(x)=a{x}^2+bx+c$ result in parabolas. The vertex is found at $x=-b/2a$.
• Square Root Functions: The domain is restricted to values where the radicand is $\ge 0$.

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Summary of Steps

1. Identify the function type: Determine if the function is linear, quadratic, or square root.
2. Create a table of values: Choose a range of $x$ values (including negative, zero, and positive values).
3. Calculate outputs: Substitute each $x$ into the function to find the corresponding $y$ or $f(x)$ value.
4. Plot the points: Mark the $(x,y)$ coordinates on a Cartesian plane.
5. Draw the curve/line: Connect the points smoothly to represent the function's behavior.