Shapes of Graphs (AQA GCSE Further Maths): Exam Questions

Here is a sketch of the curve  $y=a{b}^{-x}$ where $a$ and $b$ are positive constants.

$(0,3)$ and $(2,0.48)$ lie on the curve.

Work out the values of $a$ and $b$.

$a=$ ...............................    

$b=$ ...............................

Here is a sketch of the curve $y=(2x+3)(x–2)$

The curve intersects the $x$-axis at $A$ and $B$.

Complete the coordinates of $A$ and $B$.

$A(......,0)B(.......,0)$

Write down the range of values for $x$ for which   $(2x+3)(x–2)<0$

Here is the graph of  $y=3x-x^2$ for values of $x$ from $-1$ to $4$

By drawing a suitable linear graph on the grid, work out approximate solutions to

$x^2-4x+2=0$

Here is a sketch of  $y=a+bx-2x^2$  where $a$ and $b$ are constants.

The graph intersects the $x$-axis at $\left(-1,0\right)$ and $\left(\frac{7}{2},0\right)$ and the $y$-axis at point $P.$

Work out the coordinates of point $P$.

You must show your working.

The graph of $y=\left(x-p\right)\left(x-5p\right)$ is shown, where $p>0$.

The line $y=-1$ intersects the graph exactly once.

Find the value of $p$.

$y=\mathrm{f}\left(x\right)$ is a cubic curve with a maximum and a minimum stationary point.

$\frac{\mathrm{d}y}{\mathrm{d}x}=x^2+2x-3$

The $y$-coordinate of the minimum point is  $2\frac{1}{3}$.

The $y$-coordinate of the maximum point is  $13$.

$(0,4)$ is a point on the curve.

The tangent at $(0,4)$ has a negative gradient.

Sketch the curve on the grid below and show the coordinates of the stationary points.

The continuous curve $y=$g($x$)  has exactly two stationary points.

The stationary points are

• a maximum point at $P\left(3a,b\right)$ where $a>0$ and $b<0$

•  a minimum point at $Q\left(-a,3b\right)$

On the axes below, sketch the curve.

Label points $P$ and $Q$ on your sketch.

The graph of $y=a{b}^x$ is shown, where $a$ and $b$ are positive constants.

The graph has a $y$-intercept of $\frac{1}{4}$ and passes through the point $\left(\frac{2}{3},1\right)$

By finding the values of $a$ and $b$, work out the equation of the graph.

Give your answer in the form $y=c^{px+q}$ where $p$ and $q$ are integers and where $c$ is the smallest positive integer possible.