A-Level AQA Maths Pure Simultaneous Equations: Figure 1 shows the graph of $y =

Figure 1 shows the graph of $y = |2x|$ - AQA - A-Level Maths Pure - Question 4 - 2021 - Paper 2

Question 4

Figure 1 shows the graph of $y = |2x|$. On Figure 1 add a sketch of the graph of $y = |3x - 6|$. Find the coordinates of the points of intersection of the two gra... show full transcript

Worked Solution & Example Answer:Figure 1 shows the graph of $y = |2x|$ - AQA - A-Level Maths Pure - Question 4 - 2021 - Paper 2

Step 1

On Figure 1 add a sketch of the graph of $y = |3x - 6|$.

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Answer

To sketch the graph of the equation $y = |3x - 6|$ , we first identify the vertex. Setting the inside of the absolute value to zero gives:

$3x - 6 = 0$

Solving for $x$ yields:

$x = 2$

At $x = 2$ , the value of $y$ is:

$y = |3(2) - 6| = |0| = 0$

The vertex is at the point (2, 0). The line $y = 3x - 6$ will be positive when $x > 2$ and negative when $x < 2$ . Therefore, the graph will form a 'V' shape with its vertex at (2, 0). The graph touches the x-axis at this vertex and rises to the right and left. Make sure to clearly label the vertex and the direction of the lines.

Step 2

Find the coordinates of the points of intersection of the two graphs.

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Answer

To find the points of intersection between $y = |2x|$ and $y = |3x - 6|$ , we set the equations equal:

$|2x| = |3x - 6|$

This leads to two cases based on the definition of absolute values:

Case 1: $2x = 3x - 6$

Rearranging gives:

$2x - 3x = -6 \implies -x = -6 \implies x = 6$

Substituting $x = 6$ back into either original equation to find $y$ :

$y = |2(6)| = 12$

Thus, one point of intersection is (6, 12).

Case 2: $2x = - (3x - 6)$

Rearranging gives:

$2x = -3x + 6 \implies 2x + 3x = 6 \implies 5x = 6 \implies x = 1.2$

Substituting $x = 1.2$ back into either original equation to find $y$ :

$y = |2(1.2)| = 2.4$

Thus, the second point of intersection is (1.2, 2.4).

Summary of Intersection Points:

(6, 12)
(1.2, 2.4)