A-Level AQA Maths Pure Quadratics: A curve, C, has equation $y = x^

A curve, C, has equation $y = x^2 - 6x + k$, where $k$ is a constant - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 2

Question 4

A curve, C, has equation $y = x^2 - 6x + k$, where $k$ is a constant. The equation $x^2 - 6x + k = 0$ has two distinct positive roots. 4 (a) Sketch C on the axes b... show full transcript

Worked Solution & Example Answer:A curve, C, has equation $y = x^2 - 6x + k$, where $k$ is a constant - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 2

Step 1

Sketch C on the axes below.

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Answer

To sketch the curve defined by the equation $y = x^2 - 6x + k$ , we start by recognizing that this is a quadratic function with a standard 'U' shape.

Vertex Calculation: The vertex of a parabola defined by $y = ax^2 + bx + c$ can be found using the formula for the x-coordinate of the vertex, which is given by:

$x = -\frac{b}{2a}$

In this case, $a = 1$ and $b = -6$ , thus:

$x = -\frac{-6}{2 \cdot 1} = 3$

Finding the y-coordinate of the vertex: Substituting $x = 3$ back into the equation:

$y = (3)^2 - 6(3) + k = 9 - 18 + k = k - 9$

Intersections with Axes: The curve intersects the y-axis at $(0, k)$ and the x-axis at the points where $y = 0$ (roots of the equation) when $k$ is suitably chosen.

Graphing: Considering the vertex and the y-intercept, sketch a U-shaped curve opening upwards, ensuring it intersects the x-axis at two distinct points, represented with appropriate axis labels.

Step 2

Find the range of possible values for k.

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Answer

To find the range of possible values for $k$ , we analyze the quadratic equation $x^2 - 6x + k = 0$ .

Condition for Distinct Roots: For the quadratic to have two distinct roots, the discriminant must be greater than zero:

$D = b^2 - 4ac > 0$

Substituting $a = 1$ , $b = -6$ , and $c = k$ , we have:

$(-6)^2 - 4(1)(k) > 0$

This simplifies to:

$36 - 4k > 0$

Solving for $k$ , we get:

$36 > 4k \\ k < 9$

Condition for Positive Roots: Additionally, since we need both roots to be positive, the y-intercept $k$ must also be positive (as the curve must cross the x-axis above it). Hence, we have:

$k > 0$

Conclusion: Combining both conditions, we find that:

$0 < k < 9$

Thus, the full range of possible values for $k$ is $0 < k < 9$ .

A curve, C, has equation $y = x^2 - 6x + k$, where $k$ is a constant - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 2

Worked Solution & Example Answer:A curve, C, has equation $y = x^2 - 6x + k$, where $k$ is a constant - AQA - A-Level Maths Pure - Question 4 - 2018 - Paper 2

Sketch C on the axes below.

Find the range of possible values for k.

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