3 - Slope, intercepts, graphs

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Linear Functions and Graphs

Section 3: Practise questions

Question 1

Consider a linear function: $$y=a+bx$$ Suppose the rule of the function is such that when $𝑥$ is $60$, $𝑦$ is $66$ and when $𝑥$ is $100$, $𝑦$ is $90$. (That is, you need to consider a linear function which passes through $(60, 66)$ and ($100, 90)$. What are the values of $𝑎$ and $𝑏$ in the function? Draw the graph of the function.

The slope of the function $$b = \frac{90 - 66}{100 - 60}$$ $$b = \frac{24}{40}$$ $$b = 0.6$$

When $x=$$100$, $y=$$90$, and $b=$$0.6$ (all these numbers must satisfy the function): $$90 = a + 0.6(100)$$ $$a = 30$$

The linear function that passes through $(60, 100)$ and $(66, 90)$ is: $$y = 30 + 0.6x$$ Which is an upward-sloping line with the slope $0.6$ and the vertical intercept $(0,30)$.

Figure 5

Question 2

Draw the graph of the function: $$C = 100 + 0.8y$$ Where, $C$ is consumer spending and $Y$ is income.

Here $Y$ is income which is non-negative.
The given consumption function is a linear function with the slope $0.8$ and vertical intercept $(0,100)$. See Figure 6.

Figure 6

Question 3

Given the demand function, $$Q = 100 - 2P$$ Derive the slope of the inverse demand function. Draw the inverse demand function.

Here, $$Q = 100 - 2P$$ $$2P = 100 - Q$$ $$P = 50 - 0.5Q$$

The slope of the inverse demand function is $-0.5$, and the vertical intercept is $(0,50)$.