1. $y = ax^0 = a(1) = a$

Graph of $y = ax^0$, for $a > 0$

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Graph of $y = ax^0$, for $a < 0$

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2. $y = ax^1 = ax$

Graph of $y = ax^1$, for $a > 0$

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Graph of $y = ax^1$, for $a < 0$

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3. $y = ax^{-1} = {a \over x}$

Graph of $y = ax^{-1}$, for $a > 0$

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Note: The curve does not intersect the $y$-axis. When $x = 0$, $y = {a \over x}$ is undefined, so the graph has no point with $x = 0$.

Graph of $y = ax^{-1}$, for $a < 0$

Shape

Note: The curve does not intersect the $y$-axis. When $x = 0$, $y = {a \over x}$ is undefined, so the graph has no point with $x = 0$.

4. $y = ax^2$

Graph of $y = ax^2$, for $a > 0$

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Graph of $y = ax^2$, for $a < 0$

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5. $y = ax^{-2} = {a \over x^2}$

Graph of $y = ax^{-2}$, for $a > 0$

Shape

Note: The curve does not intersect the $y$-axis. When $x = 0$, $y = {a \over x^2}$ is undefined, so the graph has no point with $x = 0$.

Graph of $y = ax^{-2}$, for $a < 0$

Shape

Note: The curve does not intersect the $y$-axis. When $x = 0$, $y = {a \over x^2}$ is undefined, so the graph has no point with $x = 0$.

6. $y = ax^3$

Graph of $y = ax^3$, for $a > 0$

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Graph of $y = ax^3$, for $a < 0$

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1. $y = a^x$, for $a > 1$

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2. $y = a^x$, for $0 < a < 1$

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1. The point $P(1, 2)$ is marked on the axes below.

On the axes, sketch the graph of $y = \frac{2}{x^2}$.

Solutions

It does not intersect the $y$-axis because $y = \frac{2}{x^2}$ is undefined when $x = 0$.

2. On the axes below, sketch the graph of $y = 3^{-x}$. Indicate the $y$-intercept of the graph.

Solutions

\begin{align*} y & = 3^{-x} \\ y & = \frac{1}{3^x} \qquad \qquad \left[ a^{-n} = \frac{1}{a^n} \right] \\ y & = \left( \frac{1}{3} \right)^x \qquad [\text{As } x \text{ increases, } y \text{ decreases}] \\ \\ \text{Let } & x = 0, \\ y & = 3^{-(0)} \\ y & = 1 \\ \\ \implies y & \text{-intercept is } (0, 1) \end{align*}

The graph remains above the $x$-axis and gets closer to it without intersecting it.

3. The graph of $y = ka^x$, where $k$ and $a$ are real numbers, is shown below.

The graph has a $y$-intercept of $\frac{3}{2}$ and passes through the points $A(1, 6)$ and $B(c, 24)$, where $c$ is an integer.

Find the values of $k$, $a$ and $c$.

Answer: $ k = \frac{3}{2}, a = 4, c = 2 $

Solutions