A-Level Edexcel Maths Pure Rational Expressions: 10. (a) On the axes below, sketc

10. (a) On the axes below, sketch the graphs of (i) $y = x(x + 2)(3 - x)$ (ii) $y = -\frac{2}{x}$ showing clearly the coordinates of all the points where the curves cross the coordinate axes - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 2

Question 3

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10. (a) On the axes below, sketch the graphs of (i) $y = x(x + 2)(3 - x)$ (ii) $y = -\frac{2}{x}$ showing clearly the coordinates of all the points where the cu... show full transcript

Worked Solution & Example Answer:10. (a) On the axes below, sketch the graphs of (i) $y = x(x + 2)(3 - x)$ (ii) $y = -\frac{2}{x}$ showing clearly the coordinates of all the points where the curves cross the coordinate axes - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 2

Step 1

(i) $y = x(x + 2)(3 - x)$

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Answer

To graph the function, we first identify the x-intercepts and y-intercepts.

Finding x-intercepts: Set $y = 0$ :

$x(x + 2)(3 - x) = 0$

This gives us: $x = 0$ $x + 2 = 0 \Rightarrow x = -2$ $3 - x = 0 \Rightarrow x = 3$

Therefore, the x-intercepts are at $(0, 0)$ , $(-2, 0)$ , and $(3, 0)$ .
Finding y-intercept: Set $x = 0$ :

$y = 0(0 + 2)(3 - 0) = 0$

The y-intercept is at $(0, 0)$ .
Shape of the graph: Since this is a cubic function, it will cross the x-axis at three points and has a general shape of an inverted 'W'.

Step 2

(ii) $y = -\frac{2}{x}$

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Answer

To graph this function, we analyze its characteristics:

Finding x-intercepts: Set $y = 0$ :

The equation $-\frac{2}{x} = 0$ has no solutions as the function approaches 0 but never reaches it.
Finding y-intercept: Set $x = 0$ is undefined, so there is no y-intercept.
Asymptotes: The graph will have a vertical asymptote at $x = 0$ , and it will strike the curve in the second and fourth quadrants, approaching the axes but never crossing them.

Step 3

Using your sketch state, giving a reason, the number of real solutions to the equation $x(x + 2)(3 - x) + \frac{2}{x} = 0$

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Answer

From the sketch, we observe that the first function has its roots at $x = 0$ , $x = -2$ , and $x = 3$ , while the second function approaches but does not cross the x-axis.

The equation will have two real solutions since it intersects the curve of the first function at two points, confirming that there are two points of intersection where the overall equation equals zero.

10. (a) On the axes below, sketch the graphs of (i) $y = x(x + 2)(3 - x)$ (ii) $y = -\frac{2}{x}$ showing clearly the coordinates of all the points where the curves cross the coordinate axes - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 2

Worked Solution & Example Answer:10. (a) On the axes below, sketch the graphs of (i) $y = x(x + 2)(3 - x)$ (ii) $y = -\frac{2}{x}$ showing clearly the coordinates of all the points where the curves cross the coordinate axes - Edexcel - A-Level Maths Pure - Question 3 - 2011 - Paper 2

(i) $y = x(x + 2)(3 - x)$

(ii) $y = -\frac{2}{x}$

Using your sketch state, giving a reason, the number of real solutions to the equation $x(x + 2)(3 - x) + \frac{2}{x} = 0$

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