Figure 1 shows a sketch of the curve with equation $$y = \frac{2}{x}, \quad x \neq 0$$ The curve C has equation $$y = \frac{2}{x} - 5, \quad x \neq 0$$ and the line l has equation $$y = 4x + 2.$$ (a) Sketch and clearly label the graphs of C and l on a single diagram. On your diagram, show clearly the coordinates of the points where C and l cross the coordinate axes. (b) Write down the equations of the asymptotes of the curve C. (c) Find the coordinates of the points of intersection of $$y = \frac{2}{x} - 5$$ and $$y = 4x + 2.$$

A-Level Edexcel Maths Pure Modelling with Sequences & Series: Figure 1 shows a sketch of the c

Figure 1 shows a sketch of the curve with equation $$y = \frac{2}{x}, \quad x \neq 0$$ The curve C has equation $$y = \frac{2}{x} - 5, \quad x \neq 0$$ and the line l has equation $$y = 4x + 2.$$ (a) Sketch and clearly label the graphs of C and l on a single diagram - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 3

Question 7

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Figure 1 shows a sketch of the curve with equation $$y = \frac{2}{x}, \quad x \neq 0$$ The curve C has equation $$y = \frac{2}{x} - 5, \quad x \neq 0$$ and the lin... show full transcript

Worked Solution & Example Answer:Figure 1 shows a sketch of the curve with equation $$y = \frac{2}{x}, \quad x \neq 0$$ The curve C has equation $$y = \frac{2}{x} - 5, \quad x \neq 0$$ and the line l has equation $$y = 4x + 2.$$ (a) Sketch and clearly label the graphs of C and l on a single diagram - Edexcel - A-Level Maths Pure - Question 7 - 2013 - Paper 3

Step 1

Sketch and clearly label the graphs of C and l on a single diagram.

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Answer

Sketch the curve C: Start by plotting the hyperbola defined by the equation $y = \frac{2}{x}$ . The curve will approach the x-axis (y = 0) and y-axis (x = 0) as asymptotes.
Shift the hyperbola: Since C’s equation is $y = \frac{2}{x} - 5$ , shift the entire hyperbola downward by 5 units. The new asymptote is now at y = -5.
Sketch the line l: Draw the line given by the equation $y = 4x + 2$ . This line has a y-intercept at (0, 2) and a slope of 4.
Label axes crossings: Identify points where C crosses the x-axis (i.e., set y = 0) and where l crosses the axes. Ensure these points are marked clearly on the diagram.

Step 2

Write down the equations of the asymptotes of the curve C.

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Answer

The asymptotes for the curve C given by (y = \frac{2}{x} - 5) are:

The vertical asymptote at (x = 0)
The horizontal asymptote at (y = -5)

Step 3

Find the coordinates of the points of intersection of y = 2/x - 5 and y = 4x + 2.

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Answer

To find the points of intersection, set the equations equal:
$\frac{2}{x} - 5 = 4x + 2$

Multiply through by x to eliminate the fraction:
$2 - 5x = 4x^2 + 2x$
Rearranging gives:
$4x^2 + 7x - 2 = 0$
Use the quadratic formula to solve for x:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-7 \pm \sqrt{49 + 32}}{8} = \frac{-7 \pm 9}{8}$
This gives two potential solutions:
- (x = \frac{1}{4} \approx 0.25)
- (x = -2)
Now, substitute these x-values back into either original equations to obtain corresponding y-values:
- For (x = \frac{1}{4}):
  $y = 4\left(\frac{1}{4}\right) + 2 = 3$
- For (x = -2):
  $y = 4(-2) + 2 = -6$

The points of intersection are approximately (\left(\frac{1}{4}, 3\right)) and ((-2, -6)).

Sketch and clearly label the graphs of C and l on a single diagram.

Write down the equations of the asymptotes of the curve C.

Find the coordinates of the points of intersection of y = 2/x - 5 and y = 4x + 2.

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