P = {(1, a), (2, 4), (3, b), (4, c)} - Junior Cycle Mathematics - Question 15 - 2012

To draw the graph of the function, we first evaluate $f(x)$ for $x$ in the specified interval:

• For $x = -2$: $f(-2) = 5 + 2(-2) - (-2)^2 = 5 - 4 - 4 = -3$
• For $x = -1$: $f(-1) = 5 + 2(-1) - (-1)^2 = 5 - 2 - 1 = 2$
• For $x = 0$: $f(0) = 5 + 2(0) - 0^2 = 5$
• For $x = 1$: $f(1) = 5 + 2(1) - 1^2 = 5 + 2 - 1 = 6$
• For $x = 2$: $f(2) = 5 + 2(2) - 2^2 = 5 + 4 - 4 = 5$
• For $x = 3$: $f(3) = 5 + 2(3) - 3^2 = 5 + 6 - 9 = 2$
• For $x = 4$: $f(4) = 5 + 2(4) - 4^2 = 5 + 8 - 16 = -3$

Plotting these points, we can connect them to visualize the parabola.

The axis of symmetry for a quadratic function of the form $f(x) = ax^2 + bx + c$ can be found using the formula:

$x = -\frac{b}{2a}$

In our function $f(x) = -x^2 + 2x + 5$, we have $a = -1$ and $b = 2$. Thus, the axis of symmetry is:

$x = -\frac{2}{2(-1)} = 1$

So, we will draw a vertical dashed line at $x = 1$ on the graph.

Using the graph that we have drawn, we can look for the point at $x = 1.5$. The corresponding value $f(1.5)$ can be observed directly from the graph. According to our estimation from the graph, the value is approximately:

• $f(1.5) = 5.5$.