Linear Modelling and Basic Graphs

Algebra, STD2 A2 2024 HSC 2 MC

Consider the function shown.

Which of the following could be the equation of this function?

$y=2 x+3$
$y=2 x-3$
$y=-2 x+3$
$y=-2 x-3$

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$C$

Show Worked Solution

$\text {Gradient is negative (top left } \rightarrow \text { bottom right)}$

$y \text{-intercept = 3 (only positive option)}$

$\Rightarrow C$

Algebra, STD2 A2 2022 HSC 2 MC

Which of the following could be the graph of `y=-2 x+2`?

Show Answers Only

`A`

Show Worked Solution

`text{By elimination:}`

`y text{-intercept = 2 → Eliminate}\ B and C`

`text{Gradient is negative → Eliminate}\ D`

`=>A`

♦ Mean mark 48%.

Algebra, STD1 A2 2021 HSC 5 MC

Which of the following best represents the graph of `y = -3x + 4`?

A.		B.
C.		D.

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`A`

Show Worked Solution

`y = -3x + 4`

♦♦♦ Mean mark 16%.

`ytext(-intercept) = 4`

`text(Gradient is negative → slopes from)`

`text(upper left to lower right)`

`=> A`

Algebra, STD2 A2 2020 HSC 6 MC

Suppose `y = -1-2x`.

When the value of `x` increases by 5, the value of `y` decreases by

Show Answers Only

`D`

Show Worked Solution

`text(Strategy 1)`

`text(If) \ \ x\ \ text(increases by) \ 5`

`=> y \ \ text(decreases by) \ \ 2x = 2 xx 5 = 10`

`text(Strategy 2)`

`text(Test) \ 2 \ text(values:)`

`text(If) \ \ x = 0 , \ y =-1`

`text(If) \ \ x = 5 , \ y =-1-10 =-11`

`:.\ y \ text(decreases by) \ 10.`

`=>D`

Algebra, STD2 A2 EQ-Bank 12 MC

Which of these equations represents the line in the graph?

`y = 11-2x`
`y = 11 + 5.5x`
`y = 11-5.5x`
`y = 11 + 2x`

Show Answers Only

`A`

Show Worked Solution

`text{Graph passes through (0, 11) and (5.5, 0):}`

`\text{Gradient} = \text{rise}/\text{run} = -11/5.5=-2`

`text(Gradient)`	`=(y_2-y_1)/(x_2-x_1)`
	`=(11-0)/(0-5.5)`
	`=-2`

`:.\ text(Equation is:)\ \ y = 11-2x`

`=> A`

Algebra, STD2 A2 EQ-Bank 8 MC

What is the equation for the line shown?

`y = -3/2 x + 2`
`y = 3/2 x + 2`
`y = -2/3 x + 2`
`y = 2/3 x + 2`

Show Answers Only

`B`

Show Worked Solution

`text(Graph cuts)\ y text(-axis at 2)`

`text(Gradient) = text(rise)/text(run) = 3/2`

`:. y = 3/2 x + 2`

`=>B`

Algebra, STD2 A2 EQ-Bank 10 MC

Leo drew a straight line through the points `(0, 5)` and `(3, -2)` as shown in the diagram below.

What is the gradient of the line that Leo drew?

`7/3`
`3/7`
`-3/7`
`-7/3`

Show Answers Only

`D`

Show Worked Solution

`text(Line passes through)\ (0, 5)\ \text(and)\ (3,-2):`

`text(Gradient)= \text{rise}/\text{run} = (5-(-2))/(0-3)= -7/3`

`=>D`

Algebra, STD2 A2 EQ-Bank 13 MC

Renee went bike riding on a holiday.

The hiring charges are listed in the table below:

\begin{array} {|l|c|c|}
\hline \text{Hours hired} \ (h) & 1 & 2 & 3 & 4 & 5 \\
\hline \text{Cost} \ (C) & 18 & 24 & 30 & 36 & 42 \\
\hline \end{array}

Which linear equation shows the relationship between `C` and `h`?

`C = 12 + 6h`
`C = 6 + 12h`
`C=18 + 12h`
`C=12 + 18h`

Show Answers Only

`A`

Show Worked Solution

`text(Consider Option 1:)`

`12 + (6 xx 1) = 12+6=18`

`12 + (6 xx 2) = 12+12=24`

`12 + (6 xx 3) = 12+18=30\ \ \ \ text(etc …)`

`:.\ text(The linear equation is:)\ \ C = 12 + 6h`

`=>A`

Algebra, STD2 A2 EQ-Bank 17

The diagram shows the graph of a line.

Determine the equation of this line. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`y = 3/5x + 3`

Show Worked Solution

`ytext(-intercept = 3)`

`text(Gradient)\ = text(rise)/text(run)= 3/5`

`:. text(Equation:)\ \ y = 3/5x + 3`

Algebra, STD2 A2 EQ-Bank 9 MC

The equation of the line drawn in the diagram below is:

`5y = 3x`
`y = 5/3x + 5`
`y = 3/5x + 5`
`y = -5/3x + 5`

Show Answers Only

`D`

Show Worked Solution

`y text(-intercept) = +5`

`text(Gradient)= text(rise)/text(run)=-5/3`

`:.\ text(Equation is):\ y =-5/3x + 5`

`=> D`

Algebra, STD2 A2 EQ-Bank 2 MC

What is the gradient of the line `2x + 3y + 4 = 0`?

`-2/3`
`2/3`
`-3/2`
`3/2`

Show Answers Only

`A`

Show Worked Solution

`2x + 3y + 4`	`= 0`
`3y`	`= -2x-4`
`y`	`= -2/3 x-4/3`

`=> A`

Algebra, STD2 A2 EQ-Bank 1 MC

What is the slope of the line with equation `5x+2y-8 = 0`?

`-5/2`
`-2/5`
`2/5`
`5/2`

Show Answers Only

`A`

Show Worked Solution

`5x+2y-8`	`= 0`
`2y`	`= -5x+8`
`y`	`= -5/2 x+4`

`:.\ text(Slope)\ = -5/2`

`=>A`

Algebra, STD2 A2 EQ-Bank 16

Sketch the graph of `y-2x = 3`, showing the intercepts on both axes. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

Show Worked Solution

`y-2x=3\ \ => \ \ y=2x+3`

Algebra, STD2 A4 2017 HSC 17 MC

The graph of the line with equation `y = 6-2x` is shown.

When the graph of the line with equation `y = x + 3` is also drawn on this number plane, what will be the point of intersection of the two lines?

A. `(0, 6)`

B. `(1, 4)`

C. `(2, 2)`

D. `(3, 0)`

Show Answers Only

`B`

Show Worked Solution

`text(Solution 1)`

`text(From graph, intersection at (1,4))`

`=>B`

`text(Solution 2)`

`y`	`= 6 – 2x`	`…\ (1)`
`y`	`= x + 3`	`…\ (2)`

`text{Substitute (2) into (1)}`

`x + 3`	`= 6 – 2x`
`3x`	`= 3`
`x`	`= 1`

`text(When)\ \ x = 1, y = 4`

`=>B`

Algebra, STD2 A2 2016 HSC 14 MC

The graph shows a line which has an equation in the form `y = mx + c`.

Which of the following statements is true?

`m` is positive and `c` is negative
`m` is negative and `c` is positive
`m` and `c` are both positive
`m` and `c` are both negative

Show Answers Only

`=> A`

Show Worked Solution

`m` is the gradient and the line slopes to the right so `m` is positive.

`c` is the `y`-intercept which is negative.

`:.\ m` is positive and `c` is negative.

`=> A`

Algebra, STD2 A2 2015 HSC 13 MC

What is the equation of the line `l`?

`y = -2x + 2`
`y = 2x + 2`
`y = -x/2 + 2`
`y = x/2 +2`

Show Answers Only

`A`

Show Worked Solution

`l\ text{passes through (0, 2) and (1, 0)}`

♦ Mean mark 48%.

`text(Gradient)`	`= text(rise)/text(run)`
	`=-2/1`
	`= -2`

`y \text(-intercept = 2)`

`:.y = -2x + 2`

`=>A`

Algebra, STD2 A2 2005 HSC 17 MC

The total cost, `$C`, of a school excursion is given by `C = 2n + 5`, where `n` is the number of students.

If three extra students go on the excursion, by how much does the total cost increase?

`$6`
`$11`
`$15`
`$16`

Show Answers Only

`A`

Show Worked Solution

`C = 2n + 5`

`text(If)\ n\ text(increases to)\ n + 3`

`C`	`= 2(n + 3) + 5`
	`= 2n + 6 + 5`
	`= 2n + 11`

`:.\ text(Total cost increases by $6)`

`=> A`

Algebra, STD2 A2 2006 HSC 7 MC

Which equation represents the relationship between `x` and `y` in this table?

`y = 2x+1`
`y = 2x-2`
`y = x/2-2`
`y = x/2+1`

Show Answers Only

`D`

Show Worked Solution

`text{By elimination (using the table)}`

`(0, 1)\ text(must satisfy)`

`:.\ text(NOT)\ B\ text(or)\ C`

`(2, 2)\ text(must satisfy)`

`:.\ text(NOT)\ A`

`=> D`

Algebra, STD2 A2 2004 HSC 2 MC

Susan drew a graph of the height of a plant.

What is the gradient of the line?

`1`
`5`
`7.5`
`10`

Show Answers Only

`B`

Show Worked Solution

`text(2 points on graph)\ \ (0, 10),\ (1, 15)`

`text(Gradient)`	`= (y_2-y_1) / (x_2-x_1)`
	`= (15-10) / (1-0)`
	`= 5`

`=> B`

Algebra, STD2 A2 2014 HSC 7 MC

Which of the following is the graph of `y = 2x-2`?

Show Answers Only

`D`

Show Worked Solution

♦ Mean mark 46%

`y = 2x-2`

`text(From the equation:)`

`y\ text(intercept of)\ -2`

`=> text(Eliminate)\ B\ text(and)\ C`

`text(From the equation:)`

`text(Gradient)=2`

`=> text(Eliminate)\ A`

`=> D`

Algebra, STD2 A2 2009 HSC 14 MC

If `A = 6x + 10`, and `x` is increased by 2, what will be the corresponding increase in `A` ?

`2x`
`6x`
`2`
`12`

Show Answers Only

`D`

Show Worked Solution

♦ Mean mark 50%.
STRATEGY: Substituting real numbers into the equation can work well in these type of questions. eg. If `x=0,\ A=10` and when `x=2,\ A=22`.

`A = 6x + 10`

`text(If)\ x\ text(increases by 2)`

`A\ text(increases by)\ 6 xx 2 = 12`

`=> D`

Algebra, STD2 A2 2012 HSC 5 MC

The line `l` has intercepts `p` and `q`, where `p` and `q` are positive integers.

What is the gradient of line `l ` ?

`-p/q`
`-q/p`
`p/q`
`q/p`

Show Answers Only

`A`

Show Worked Solution

♦ Mean mark 45%

`text(Gradient)`	`= text(rise)/text(run)`
	`= -p/q`

`=> A`