Past Papers’ Solutions  Edexcel  AS & A level  Mathematics  Core Mathematics 1 (C16663/01)  Year 2013  January  Q#10
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Question
a. Find the values of the constants a, b and c.
b. On the axes on page 27, sketch the curve with equation y = 4x^{2 }+ 8x + 3, showing clearly the coordinates of any points where the curve crosses the coordinate axes.
Solution
a.
We are given that;
In order to find the values of the constants a, b and c, we need to write the left hand side of the equation in the same form as right hand side of the equation.
We use method of “completing square” to obtain the desired form.
Next we complete the square for the terms which involve .
We have the algebraic formula;
For the given case we can compare the given terms with the formula as below;
Therefore we can deduce that;
Hence we can write;
To complete the square we can add and subtract the deduced value of ;
Hence;
Therefore;
b.
We are required to sketch the curve with given equation;
It is evident that it is a quadratic equation.
To sketch a quadratic equation, a parabola, we need the coordinates of its vertex and x and y intercepts, if any.
First we find the coordinates of vertex of this parabola.
Standard form of quadratic equation is;
The graph of quadratic equation is a parabola. If (‘a’ is positive) then parabola opens upwards and its vertex is the minimum point on the graph.
If (‘a’ is negative) then parabola opens downwards and its vertex is the maximum point on the graph.
We recognize that given curve , is a parabola opening upwards.
Vertex form of a quadratic equation is;
The given curve
Coordinates of the vertex are
For the given case, vertex is
Next, we need x and yintercepts of the parabola.
First we find the xintercept of the parabola.
The point
Therefore, we substitute
Now we have two options.






Therefore, there are two xintercepts of the given parabolic curve with coordinates
Next, we find the yintercept of the parabola.
The point
Therefore, we substitute
Hence, coordinates of yintercept of the parabola are
We can sketch the parabola as shown below.
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