Use the symmetry of the graph to show that the second solution is $x=4$.

Find $g^{\prime }(-1)$.

Find the value of $t$ at which the goat is eating grass at the greatest rate.

Sieun observes that when the angle is $40°$, the ball will travel a horizontal distance of $205.5 \text{m}$.

Find an expression for the function $l\left(\theta \right)$.

Juri starts at a height of $60$ metres and finishes at $\text{F}$, where $x=f$.

Sketch a possible diagram of the hill on the following pair of coordinate axes.

Determine whether the profit is increasing or decreasing when $t=2$.

Find the value of $t$ at which the goat is eating grass at the greatest rate.

Write down the number of possible solutions to the equation $f(x)=5$.

Given that y = 2x³ − 9x² + 12x + 2 = k has three solutions, find the possible values of k.

Sketch the curve for −1 < x < 3 and −2 < y < 12.

Find $f^{\prime }(x)$.

Write down the $x$-coordinates of these two points;

A teacher asks her students to make some observations about the curve.

Three students responded.
Nadia said “The x-intercept of the curve is between −1 and zero”.
Rick said “The curve is decreasing when x < 1 ”.
Paula said “The gradient of the curve is less than zero between x = 1 and x = 2 ”.

State the name of the student who made an incorrect observation.

Write down the $y$-intercept of the graph.

Hence justify that $g$ is decreasing at $x=-1$.

Find $f(2)$.

Write down the equation for the axis of symmetry of the graph.

Write down the $x$-intercepts of the graph.

Draw the tangent $T$ on your graph.

Show that $a=8$.

Expand the expression for $f(x)$.

Show that $k=-6$.

Find the equation of the tangent to the graph of $y=g(x)$ at $x=2$. Give your answer in the form $y=mx+c$.

Write down the coordinates of the point of intersection.

On graph paper, draw the graph of $y=f\left(x\right)$ for  $-10\le x\le 4$  and  $-14\le y\le 14$. Use a scale of $1 \text{cm}$ to represent $1$ unit on the $x$-axis and $1 \text{cm}$ to represent $2$ units on the $y$-axis.

Write down the intervals where the gradient of the graph of $y=f(x)$ is positive.

Write down the equation of $T$.

The equation $f(x)=m$, where $m\in \mathbb{R}$, has four solutions. Find the possible values of $m$.

Use your answer to part (b)(ii) to find the values of $x$ for which $f$ is increasing.

Find $\frac{\text{dy}}{\text{dx}}$.

Find $f^{\prime }(x)$.

Show that $k=6$.

Find $g^{\prime }(x)$.

Write down the range of $f(x)$.

Find the coordinates of the local minimum.

Find the $y$-coordinate of the local minimum.

Draw the graph of $f$ for $-3⩽x⩽3$ and $-40⩽y⩽20$. Use a scale of 2 cm to represent 1 unit on the $x$-axis and 1 cm to represent 5 units on the $y$-axis.

Determine the equation of the normal at $x=-2$ in the form $y=mx+c$.

Find the value of $c$.

Determine whether the graph of $l$ against $\theta$ is increasing or decreasing at $\theta =50°$.

Use your answer to part (a) and the value of $k$, to find the $x$-coordinates of the stationary points of the graph of $y=g(x)$.

Given $f\left(a\right)=5.5$ and $f\text{'}\left(a\right)=-6$, state whether the function, $f$, is increasing or decreasing at $x=a$. Give a reason for your answer.

Write down the interval where the gradient of the graph of $f\left(x\right)$ is negative.

Identify the $x$ value of the point where $|h\prime (x\left)\right|$ has its maximum value.

Interpret this point in the given context.