Cubic, reciprocal and exponential graphs questions with answers

Questions

1. Question 1Table of values: cubic

   Given that $y = x^{3} + x^{2} - 2x$, calculate the value of $y$ when $x = 0.5$, correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Substitute the given value of x and work it out$y = (0.5)^{3} + (0.5)^{2} - 2(0.5)$$\phantom{y} = -0.625$
   2. Step 2: Round to 1 decimal place$y = -0.6$
2. Question 2Table of values: reciprocal

   Given that $y = \dfrac{4}{x}$, calculate the value of $y$ when $x = 1.5$, correct to 1 decimal place.

   Show the worked solution

   1. Step 1: Substitute the given value of x and work it out$y = \dfrac{4}{1.5}$$\phantom{y} = 2.6666\ldots$
   2. Step 2: Round to 1 decimal place$y = 2.7$
3. Question 3Table of values: exponential

   Given that $y = 3(2^{x})$, calculate the value of $y$ when $x = 2$.

   Show the worked solution

   1. Step 1: Substitute the given value of x and work it out$y = 3 \times 2^{2}$$\phantom{y} = 12$
4. Question 4Find k from a point on the curve

   The curve $y = x^{3} + kx$ passes through the point $(-1, 4)$. Find the value of $k$.

   Show the worked solution

   1. Step 1: Substitute the point into the equation$4 = (-1)^{3} + k(-1)$$4 = -1 - k$
   2. Step 2: Solve for k$-k = 5$$k = -5$
5. Question 5Real world: exponential growth

   The number of followers of a new account after $t$ weeks is $N = 400(3^{t})$. Find the value of $t$ when there are $32\,400$ followers.

   Show the worked solution

   1. Step 1: Put in the given value of N$32400 = 400(3^{t})$
   2. Step 2: Divide both sides by the starting value$3^{t} = 81$
   3. Step 3: Write it as a power of the same base$3^{t} = 3^{4}$$t = 4$
6. Question 6Which graph? Match the equation and the shape

   Which diagram, A, B, C or D, shows the graph of $y = -\dfrac{1}{x^{2}}$?

   Show the worked solution

   1. Step 1: What type of graph is it?$y = -\dfrac{1}{x^{2}}$ is reciprocal ($\dfrac{a}{x^{2}}$).
   2. Step 2: Which way does it go?$y = -\dfrac{1}{x^{2}}$: both branches below the x-axis.
   3. Step 3: AnswerDiagram $A$: $y = -x^{2} - 1$Diagram $B$: $y = -\dfrac{1}{x}$Diagram $C$: $y = \dfrac{1}{x^{2}}$Diagram $D$: $y = -\dfrac{1}{x^{2}}$Diagram $D$
7. Question 7Real world: the depth of water against time

   Water is poured at a constant rate into the empty container shown, which is open at the top. Which graph, A, B, C or D, shows how the depth of the water, $h$ cm, changes with time, $t$ seconds? (Each graph has $h$ going up and $t$ going across.)

   Show the worked solution

   1. Step 1: How does the container's width change going up?The container gets narrower going up.
   2. Step 2: The same volume goes in every second: where it is narrow the depth rises quickly (steep), where it is wide slowly (gentle). So the graph isSo the graph is a curve rising slowly then more quickly.
   3. Step 3: AnswerGraph $A$: two straight pieces gentle then steep.Graph $B$: a curve rising quickly then more slowly.Graph $C$: a straight line.Graph $D$: a curve rising slowly then more quickly — this one.Graph $D$