A-Level AQA Maths Mechanics Kinematics Graphs: 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 Mechanics - 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 Mechanics - Question 4 - 2018 - Paper 2

Step 1

Sketch C on the axes below.

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Answer

To sketch the curve $C$ , we first recognize that the equation represents a quadratic function in standard form, $y = ax^2 + bx + c$ , where:

$a = 1$ , indicating the parabola opens upwards.
The vertex of the parabola can be found using the formula $x_v = -\frac{b}{2a}$ . Here, $b = -6$ , so:

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

Substituting $x_v = 3$ back into the equation to find the vertex's $y$ -coordinate:

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

This means that the vertex of the parabola is at $(3, k - 9)$ . Since $k$ is a constant, the parabola will be below or above the x-axis depending on its value.

Next, we determine the points where the curve intersects the y-axis, which occurs when $x = 0$ :

$y = 0^2 - 6(0) + k = k$

Thus, the y-intercept is $k$ . The sketch should be a parabola crossing the y-axis at $(0, k)$ and with a vertex at $(3, k - 9)$ .

Step 2

Find the range of possible values for k.

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Answer

To find the range of possible values for $k$ such that the equation $x^2 - 6x + k = 0$ has two distinct positive roots, we utilize the discriminant condition for distinct roots, given by:

$D = b^2 - 4ac$

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

$D = (-6)^2 - 4(1)(k) = 36 - 4k$

For the roots to be distinct, we need $D > 0$ :

ightarrow 36 > 4k\ ightarrow k < 9$$ Next, we need to ensure that the roots are positive. The vertex of the parabola at $x = 3$ gives us an insight into when the curve intersects the x-axis. The minimum value occurs at $(3, k - 9)$. For the roots to be positive, the vertex must be above the x-axis: $$k - 9 > 0\ ightarrow k > 9$$ However, this is contradictory, thus we reconsider the implications: The parabola intersects the y-axis at $(0, k)$ and since it opens upwards, we require the intersection point with the x-axis to occur before the vertex at $x = 3$. This translates to: The constraints combined give: $$0 < k < 9$$ Thus, the range for $k$ ensuring two distinct positive roots is: $$0 < k < 9$$

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

Sketch C on the axes below.

Find the range of possible values for k.

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