2.8 Transformations of graphs questions for IB Maths AI

The graph of $y=g(x)$, where $g(x)={\mathrm{e}}^x-4$, is obtained from the graph of $y=f(x)$ by a single transformation. Describe this transformation fully.

Write down the equation of the horizontal asymptote of the graph of $y=g(x)$.

The graph of $y=h(x)$, where $h(x)={\mathrm{e}}^{\frac{x}{2}}$, is obtained from the graph of $y=f(x)$ by a single transformation. Describe this transformation fully.

Write down the coordinates of the maximum point of the graph of $y=f(x)$.

Write down an integral for the area of the logo, and find this area.

For a banner, the graph of $f$ is transformed by a horizontal stretch with scale factor 1.5 and a vertical stretch with scale factor 2, to give the graph of $y=g(x)$. Find an expression for $g(x)$ and state its domain.

Find the area of the banner logo.

Show that the area of the banner logo is three times the area of the original logo, and explain this result in terms of the two stretches.

On the banner, the logo is moved so that its highest point is at $(5,8)$. Describe fully the transformation required and find the equation of the upper edge of the logo in its new position.

Let $g(x)=3f(x+1)-4$. Find the coordinates of the local maximum point on the graph of $y=g(x)$.

Let $h(x)=-f(2x)$. Find the coordinates of the image of the point $(2,5)$ on the graph of $y=h(x)$, and state whether it is a local maximum or a local minimum point of $h$.

Write down the maximum depth of water at Port A and the time at which it occurs.

Find the period of $D_A$.

At Port B, further along the coast, high tide occurs 1.5 hours later than at Port A, the difference between high and low tide is 1.4 times that at Port A, and the mean depth is 2 m less than at Port A.

Find an expression for $D_B(t)$, the depth of water at Port B, in the form $a\cos (b(t-c))+d$.

Describe a sequence of transformations that maps the graph of $D_A$ onto the graph of $D_B$.

A cargo ship can enter Port B only when the depth of water is at least 5 m. Find the total length of time between midnight and noon during which the ship can enter Port B.

Find the first time after midnight at which the depth of water is the same at both ports.

Find an expression for $g(x)$.

Find $g(4)$.

Kofi applies the same two transformations to the graph of $y=f(x)$, but in the opposite order. Find the value of Kofi's function when $x=4$.

Find the equilibrium quantity and price.

The government introduces a tax of 6 CHF on each pair of boots sold. As a result, the graph of the supply curve is translated by the vector $\left(\begin{array}{c}0\\ 6\end{array}\right)$. Write down the equation of the new supply curve.

Find the new equilibrium quantity and price.

Find the increase in the equilibrium price caused by the tax.

Later, an advertising campaign means that customers will buy 30 thousand more pairs at every price. Describe fully the transformation of the demand curve, and write down the equation of the new demand curve in the form $p=a-0.4q$.

Find the equilibrium price after the advertising campaign, given that the tax is still in place.