Question Statement

Plot the graphs of the following functions:

(i) $f(x)=-x^2+1$

(ii) $f(x)=2x^3$

(iii) $f(x)=1+x^{-2}=1+\frac{1}{x^2}$

(iv) $f(x)=3\sqrt{x}$

(v) $f(x)=2-x^{-1/2}=2-\frac{1}{\sqrt{x}}$

(vi) $f(x)=x^{2.5}$

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Background and Explanation

To plot a function, we evaluate the expression for various values of $x$ to create a table of coordinates $(x,f(x))$. These points are then plotted on a Cartesian plane and connected with a smooth curve to represent the function's behavior, such as symmetry, intercepts, and asymptotes.

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Solution

Part (i) $f(x)=-x^2+1$

This is a quadratic function. Since the coefficient of $x^2$ is negative, the parabola opens downwards. The $+1$ shifts the vertex to $(0,1)$.

Part (ii) $f(x)=2x^3$

This is a cubic function scaled by a factor of 2. It passes through the origin and increases rapidly for positive $x$ and decreases rapidly for negative $x$.

Part (iii) $f(x)=1+\frac{1}{x^2}$

This function is undefined at $x=0$ (vertical asymptote). As $x$ becomes very large or very small, the term $\frac{1}{x^2}$ approaches 0, meaning the graph has a horizontal asymptote at $y=1$.

Part (iv) $f(x)=3\sqrt{x}$

This is a square root function. It is only defined for $x\ge 0$. The values grow slower than a linear function but continue to increase indefinitely.

Part (v) $f(x)=2-\frac{1}{\sqrt{x}}$

As $x$ approaches 0 from the right, $f(x)$ approaches $-\infty$. As $x$ approaches infinity, $f(x)$ approaches the horizontal asymptote $y=2$.

Part (vi) $f(x)=x^{2.5}$

This is a power function where $f(x)=x^{5/2}$. It grows faster than a quadratic function ($x^2$) but slower than a cubic function ($x^3$).

(Calculation check: $4^{2.5}=(4^{1/2}{)}^5=2^5=32$; $9^{2.5}=(9^{1/2}{)}^5=3^5=243$)

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Key Formulas or Methods Used

• Point-Plotting Method: Selecting $x$ values, calculating $y$, and plotting $(x,y)$.
• Power Rules:
  • $x^{-n}=\frac{1}{x^n}$
  • $x^{1/2}=\sqrt{x}$
  • $x^{a/b}=\sqrt[b]{x^a}$
• Asymptotes: Identifying values where the function is undefined or approaches a constant.

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Summary of Steps

1. Identify the Domain: Determine which $x$ values are valid (e.g., $x\ge 0$ for square roots).
2. Create a Table: Choose a range of $x$ values (including negative, zero, and positive where applicable).
3. Calculate $f(x)$: Plug each $x$ into the function to find the corresponding $y$ value.
4. Plot Points: Mark the $(x,y)$ coordinates on a graph.
5. Draw the Curve: Connect the points smoothly, paying attention to asymptotes and end behavior.