Complete the table for the relation y = 2x + 3 − x2. Draw the graph of y = 2x + 3 − x2 and y − x = 0.

We substitute each given $x$ value into the equation $y = 2x + 3 - x^2$ to find the corresponding $y$ values.

• For $x = -2$: $y = 2(-2) + 3 - (-2)^2 = -4 + 3 - 4 = -5$
• For $x = 1$: $y = 2(1) + 3 - (1)^2 = 2 + 3 - 1 = 4$
• For $x = 1.5$: $y = 2(1.5) + 3 - (1.5)^2 = 3 + 3 - 2.25 = 3.75$
• For $x = 2$: $y = 2(2) + 3 - (2)^2 = 4 + 3 - 4 = 3$
• For $x = 3$: $y = 2(3) + 3 - (3)^2 = 6 + 3 - 9 = 0$
• For $x = 3.5$: $y = 2(3.5) + 3 - (3.5)^2 = 7 + 3 - 12.25 = -2.25$

The completed table is:

To draw the graph, plot the points from the completed table in Step 1 on a graph paper. Use a scale of 2cm to 1 unit on both the x-axis and y-axis. Connect the points with a smooth curve to form the parabola.
The coordinates to plot are:
$(-2, -5), (-1, 0), (0, 3), (1, 4), (1.5, 3.75), (2, 3), (2.5, 1.75), (3, 0), (3.5, -2.25), (4, -5)$.

The equation $y - x = 0$ simplifies to $y = x$. This is a straight line.
To draw this line on the same axes, plot at least two points and connect them. For example:

• For $x = -2$: $y = -2$. Point: $(-2, -2)$
• For $x = 0$: $y = 0$. Point: $(0, 0)$
• For $x = 4$: $y = 4$. Point: $(4, 4)$ Draw a straight line passing through these points.

Rearrange the given equation to match the forms of the graphs:
$3 + 2x = x^2 +