Sketch the graph of $y = 5 - (x - 2)^2$, showing clearly the turning point and the y-intercept. Using your graph, explain why $5 - (x - 2)^2 = 7$ will not have a solution. Answer (space for working and answer)

1. Identify the turning point and shape of the graph

$y=5-(x-2)^2=-(x-2)^2+5$
This is in the form $y=-(x-p)^2+q$, so the turning point is $(p,q)=(2,5)$.

Since the coefficient of $(x-2)^2$ is negative, the parabola opens downwards.

2. Find the y-intercept

At the y-intercept, $x=0$.

$y=5-(0-2)^2$
$y=5-4=1$

So the y-intercept is $(0,1)$.

3. Complete the sketch with labels

Draw a smooth parabola opening downwards, with maximum point labelled $(2,5)$ and crossing the y-axis at $(0,1)$. The graph is symmetric about $x=2$.

1. Compare the required value $y=7$ with the maximum value of the graph

From the graph, the turning point is $(2,5)$ and the parabola opens downwards, so the maximum value of $y$ is $5$.

Since $7>5$, the horizontal line $y=7$ does not meet the graph of $y=5-(x-2)^2$.