The graph of a linear polynomial p(x) passes through the points (1, 5) and (3, 11).
(i) Find the polynomial p(x).
(ii) Find the coordinates where the graph of p(x) cuts the axes.
(iii) Draw the graph of p(x) and verify your answers.

Question

KnowledgeBoat · Accepted Answer

(i) Let the linear polynomial be p(x) = ax + b.

Since the graph passes through (1, 5):

5 = a(1) + b

⇒ a + b = 5 …(i)

Since the graph passes through (3, 11):

11 = a(3) + b

⇒ 3a + b = 11 …(ii)

Subtracting equation (i) from equation (ii):

(3a + b) - (a + b) = 11 - 5

⇒ 2a = 6

⇒ a = 3

Substituting a = 3 in equation (i):

3 + b = 5

⇒ b = 2

∴ The polynomial is p(x) = 3x + 2.

(ii) To find where the graph cuts the y-axis, put x = 0:

p(0) = 3(0) + 2 = 2

So, the graph cuts the y-axis at (0, 2).

To find where the graph cuts the x-axis, put p(x) = 0 (i.e., y = 0):

3x + 2 = 0

⇒ 3x = -2

⇒ x = $-\dfrac{2}{3}$

So, the graph cuts the x-axis at $\left(-\dfrac{2}{3}, 0\right)$ .

∴ The graph cuts the y-axis at (0, 2) and the x-axis at $\left(-\dfrac{2}{3}, 0\right)$ .

(iii) To draw the graph, we use the points already found: (1, 5), (3, 11), (0, 2) and $\left(-\dfrac{2}{3}, 0\right)$ .

The graph passes through all the four points, verifying our answers.

Mathematics