Here are the solutions to your questions: 2) Let the function be defined by $f(x) = 2x+1$. Complete the table below. The table is already completed correctly based on the function $f(x) = 2x+1$. For example, when $x = -2$, $f(-2) = 2(-2) + 1 = -4 + 1 = -3$. When $x = 2$, $f(2) = 2(2) + 1 = 4 + 1 = 5$. $$ \begin{array}{|c|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & -3 & -1 & 1 & 3 & 5 \\ \hline \end{array} $$ 3) Find the length of a line joining the points a) $A=(1,3)$ and $B=(4,7)$ using the distance formula. Step 1: Recall the distance formula. The distance $D$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $$D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ Step 2: Identify the coordinates of points A and B. $A = (1,3) \implies x_1 = 1, y_1 = 3$ $B = (4,7) \implies x_2 = 4, y_2 = 7$ Step 3: Substitute the coordinates into the distance formula. $$D = \sqrt{(4 - 1)^2 + (7 - 3)^2}$$ Step 4: Simplify the expression. $$D = \sqrt{(3)^2 + (4)^2}$$ $$D = \sqrt{9 + 16}$$ $$D = \sqrt{25}$$ $$D = 5$$ The length of the line joining points A and B is 5 units. The final answer is $\boxed{5}$. 4) A function is given by $y = -2x+3$, plot and find the gradient of $|AC|$ and $|BE|$. Step 1: Complete the table for the function $y = -2x+3$. • For A ($x = -2$): $y = -2(-2) + 3 = 4 + 3 = 7$. So $A=(-2, 7)$. • For B ($x = -1$): $y = -2(-1) + 3 = 2 + 3 = 5$. So $B=(-1, 5)$. • For C ($x = 0$): $y = -2(0) + 3 = 0 + 3 = 3$. So $C=(0, 3)$. • For D ($x = 1$): $y = -2(1) + 3 = -2 + 3 = 1$. So $D=(1, 1)$. • For E ($x = 2$): $y = -2(2) + 3 = -4 + 3 = -1$. So $E=(2, -1)$. The completed table is: $$ \begin{array}{|c|c|c|c|c|c|} \hline \text{Point} & A & B & C & D & E \\ \hline x & -2 & -1 & 0 & 1 & 2 \\ \hline y & 7 & 5 & 3 & 1 & -1 \\ \hline \end{array} $$ Step 2: Plot the function. To plot the function $y = -2x+3$, you would plot the points calculated above: $(-2, 7)$, $(-1, 5)$, $(0, 3)$, $(1, 1)$, and $(2, -1)$ on a Cartesian plane. Since this is a linear function, connecting these points will form a straight line. Step 3: Find the gradient of $|AC|$. The gradient $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. For points $A=(-2, 7)$ and $C=(0, 3)$: $$m_{AC} = \frac{3 - 7}{0 - (-2)} = \frac{-4}{0 + 2} = \frac{-4}{2} = -2$$ The gradient of $|AC|$ is $\boxed{-2}$. Step 4: Find the gradient of $|BE|$. For points $B=(-1, 5)$ and $E=(2, -1)$: $$m_{BE} = \frac{-1 - 5}{2 - (-1)} = \frac{-6}{2 + 1} = \frac{-6}{3} = -2$$ The gradient of $|BE|$ is $\boxed{-2}$. 5) There are how quadrants and Coordinates on the Cartesian Plane. There are 4 quadrants on the Cartesian plane. Coordinates are a set of values (typically an ordered pair $(x,y)$) that specify the exact location of a point on the Cartesian plane.