Exponential functions and their graphsEdexcel International A Level Maths: Subtopic test

1. 1
   The curve $C$ has equation $y=a^x$, where $a$ is a positive constant with $a\neq1$. $C$ passes through the point $(2,9)$.

   (a)

   Find the value of $a$.

   [1 mark]

   • A$-3$
   • B$3$
   • C$\frac92$
   • D$81$

   (b)

   Find the $y$-coordinate of the point on $C$ where $x=-1$.

   [1 mark]

   • A$-3$
   • B$3$
   • C$\frac19$
   • D$\frac13$

   (c)

   Find the $x$-coordinate of the point on $C$ with $y$-coordinate $27$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The curve $D$ has equation $y=\left(\frac25\right)^x$.

   (a)

   Which statement about $D$ is correct?

   [1 mark]

   • A$y$ decreases as $x$ increases, and $y>0$ for every $x$
   • B$y$ increases as $x$ increases, because the index $x$ is positive
   • C$D$ crosses the $x$-axis at $x=1$
   • D$D$ passes through the origin

   (b)

   Find the equation of the reflection of $D$ in the $y$-axis.

   [1 mark]

   • A$y=-\left(\frac25\right)^x$
   • B$y=-\left(\frac52\right)^x$
   • C$y=\left(\frac52\right)^x$
   • D$y=\left(\frac25\right)^{x+1}$

   (c)

   Explain why $D$ does not meet the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $y=a^x$, where $a$ is a positive constant and $a\neq1$, passes through the points $\left(-2,\frac1{25}\right)$ and $P(k,125)$.

   (a)

   Find the value of $a$.

   [3 marks]

   (b)

   Find the value of $k$. Hence solve $a^{2x-1}=a^{2k}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student sketches the curves $C_1$: $y=4^x$ and $C_2$: $y=\left(\frac14\right)^x$ on the same axes, together with the line $y=16$.

   (a)

   (i) Write down the coordinates of the point where both curves cross the $y$-axis.
   (ii) Show that this is the only point where $C_1$ and $C_2$ meet.
   (iii) Find the $x$-coordinates of the points where the line $y=16$ meets $C_1$ and $C_2$.

   [6 marks]

   (b)

   The student claims that the curve $y=4^x$ lies above the curve $y=3^x$ for every $x>0$ and below it for every $x<0$. Evaluate the claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions