Why is a=1 excluded from y=a^x?

It gives y=1 for all x, a horizontal line, not an exponential curve.

Why must a0 in y=a^x?

A negative base is undefined for many real indices, such as (-2)^{\frac12}.

Behaviour of y=a^x when a1?

Increasing (growth); y\to0 as x\to-\infty.

Behaviour of y=a^x when 0<a<1?

Decreasing (decay); y\to0 as x\to\infty.

Exponential functions and their graphsEdexcel International A Level Maths: Flashcards

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What form does an exponential function take?

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What form does an exponential function take?: $y=a^x$ with $a>0$ and $a\neq1$ .
Why is $a=1$ excluded from $y=a^x$ ?: It gives $y=1$ for all $x$ , a horizontal line, not an exponential curve.
Why must $a>0$ in $y=a^x$ ?: A negative base is undefined for many real indices, such as $(-2)^{\frac12}$ .
Where does $y=a^x$ cross the $y$ -axis?: At $(0,1)$ , because $a^0=1$ .
What is the range of $y=a^x$ ?: $y>0$ .
What is the asymptote of $y=a^x$ ?: The $x$ -axis, $y=0$ .
Does $y=a^x$ have an $x$ -intercept?: No, because $a^x>0$ for all $x$ .
Behaviour of $y=a^x$ when $a>1$ ?: Increasing (growth); $y\to0$ as $x\to-\infty$ .
Behaviour of $y=a^x$ when $0<a<1$ ?: Decreasing (decay); $y\to0$ as $x\to\infty$ .
Which transformation maps $y=a^x$ onto $y=\left(\frac1a\right)^x$ ?: Reflection in the $y$ -axis.
Point of $y=a^x$ at $x=1$ ?: $(1,a)$ .
Solve $3^x=27$ .: $3^x=3^3$ , so $x=3$ .
Value of $2^{-3}$ ?: $\frac18$ .

Exam questions on Exponential functions and their graphs

The curve $C$ has equation $y=a^x$ , where $a$ is a positive constant with $a\neq1$ . $C$ passes through the point $(2,9)$ .
Find the $x$ -coordinate of the point on $C$ with $y$ -coordinate $27$ .2 marks
The curve $D$ has equation $y=\left(\frac25\right)^x$ .
Explain why $D$ does not meet the $x$ -axis.2 marks
The curve $y=a^x$ , where $a$ is a positive constant and $a\neq1$ , passes through the points $\left(-2,\frac1{25}\right)$ and $P(k,125)$ .
Find the value of $a$ .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).