The curve $y = f(x)$ is shown on the diagram. The equation of the tangent to the curve at point $T (-1, 6)$ is $y = x + 7$. At a point $R$, another tangent parallel to the tangent at $T$ is drawn. The gradient function of the curve is given by \[ \frac{dy}{dx} = 3x^2 - 6x - 8. \] Find the coordinates of $R$.

SSCE HSC Mathematics Advanced Absolute value functions: The curve $y = f(x)$ is shown on

The curve $y = f(x)$ is shown on the diagram - HSC - SSCE Mathematics Advanced - Question 28 - 2023 - Paper 1

Question 28

The curve $y = f(x)$ is shown on the diagram. The equation of the tangent to the curve at point $T (-1, 6)$ is $y = x + 7$. At a point $R$, another tangent parallel ... show full transcript

Worked Solution & Example Answer:The curve $y = f(x)$ is shown on the diagram - HSC - SSCE Mathematics Advanced - Question 28 - 2023 - Paper 1

Step 1

Find the x-coordinate of R

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Answer

To find the x-coordinate of point $R$ , we need to set the gradient function equal to the slope of the tangent line at point $T$ .

The slope of the tangent line at $T$ is 1, so we set:

$\frac{dy}{dx} = 1$

Substituting the gradient function:

$3x^2 - 6x - 8 = 1$

Solving this gives:

$3x^2 - 6x - 9 = 0 \Rightarrow x^2 - 2x - 3 = 0$

Factoring the quadratic, we get:

$(x - 3)(x + 1) = 0$

Thus, $x = 3$ or $x = -1$ . Since point $T$ is at $(-1, 6)$ , we discard $-1$ and take:

The x-coordinate of $R$ is $3$ .

Step 2

Find the y-coordinate of R

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Answer

Now we substitute $x = 3$ back into the equation of the curve to find the y-coordinate of point $R$ . We will use the original gradient function:

$y = x^3 - 3x^2 - 8x + k$

To find $k$ , we can substitute the known point $T (-1, 6)$ :

$6 = (-1)^3 - 3(-1)^2 - 8(-1) + k$

This simplifies to:

$6 = -1 - 3 + 8 + k \Rightarrow 6 = 4 + k \Rightarrow k = 2$

So the equation of the curve is:

$y = x^3 - 3x^2 - 8x + 2$

Now substituting $x = 3$ :

$y = (3)^3 - 3(3)^2 - 8(3) + 2$

Calculating gives:

$y = 27 - 27 - 24 + 2 = -22$

Thus, the coordinates of $R$ are (3, -22).

The curve $y = f(x)$ is shown on the diagram - HSC - SSCE Mathematics Advanced - Question 28 - 2023 - Paper 1

Worked Solution & Example Answer:The curve $y = f(x)$ is shown on the diagram - HSC - SSCE Mathematics Advanced - Question 28 - 2023 - Paper 1

Find the x-coordinate of R

Find the y-coordinate of R

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