6 - Practise questions

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Linear Functions and Graphs

Section 6: More practice questions

Question 1

Solve the following quadratic equation using the quadratic formula: $$5x^{2} + 23x + 12 = 0$$

$$x = \frac{-23 \pm \sqrt{23^{2} - 4\cdot 5\cdot 12}}{2\cdot 5}$$ $$x = \frac{-23 \pm \sqrt{289}}{10}$$ $$x = \frac{(-23 \pm 17)}{10}$$ $$x = -4 \qquad x = -0.6$$

Question 2

Which of the following equations are not functions?

$y = 10 + x^{2}$
$y = 10 - 2x$
$y = \pm \sqrt{x}$
$x^{2} + y^{2} = 81$

$c$ and $d$.

$y = \pm \sqrt{x}$ is not a function because for each value of $x$, we get two values of $y$. Therefore, $y = \pm \sqrt{x}$ shows a relation only.
$x^{2} + y^{2} = 81$ is not a function. $$x^{2} + y^{2} = 81$$ $$y^{2} = 81 - x^{2}$$ $$y = \pm \sqrt{81 - x^{2}}$$ Therefore, for any $-9 \leq x < 9$, we get two values for $y$ which violates the properties of a function.

Question 3

What is the domain of each of the following two functions? $$(i) \quad f(x) = 10 - 2x \qquad\qquad (ii) \quad f(x) = \frac{x}{x^{2} - 1}$$

(i) All $x$
(ii) All real numbers except {$x = \pm 1$}

Question 4

Suppose the demand functions is $Q_{d} = 10 - 2P$, where $Q_{d}$ denotes quantity demanded and $P$ is the price. What is the domain of the function?

$R_{+} \equiv \{P \in R^{1}: P \geq 0\}$

Domain is the set of all possible values of the independent variable. Here, the independent variable is the price $P$. Price of the good cannot be negative but can be zero and positive.

Question 5

Suppose the yearly supply of wheat in Canada is: $$Q^{s} = 0.15 + P$$

Here, $Q^s$ is the quantity of wheat produced in Canada per year (in millions of bushels) when $P$ is the average price of wheat (in dollars per bushel). What is the quantity of wheat supplied per year when the average price of wheat is $$2$ per bushel?

2.15 million bushels [For more similar examples, see Besanko D and R Braeutigam, Microeconomics, Wiley.]

Question 6

To draw the graph of the following equation, find the slope and vertical intercept if $y$ is measured on the vertical axis: $$8y + 2x + 24 = 0$$

$$8y + 2x + 24 = 0$$ $$\qquad\qquad\qquad\qquad 8y = -24 - 2x$$ $$\qquad\qquad\qquad\qquad y = -3 - \frac{2}{8}x$$

Slope $m = -\frac{1}{4}$, vertical intercept: $(0,-3)$

Question 7

Find the formula for the linear function whose graph has slope $4$ and goes through the point $(1, 1)$.

Linear function $$f(x) = a + bx$$

If its graph has slope $4$ and goes through the point $(1,1)$, these numbers must satisfy the equation: $$1 = a + 4\cdot 1$$ $$a = -3$$ So, the function is $$f(x) = 4x - 3$$