Key features and models of graphs Problem set B

The function \(g(x)=-x^2+8x-7\) models a profit. Use your calculator to find the y-intercept, the zeros, the vertex and the equation of the axis of symmetry.

Write down the equation of the horizontal asymptote of each graph.

1. (a) \(y=2\times3^x-6\)
2. (b) \(y=4\times0.8^x+1\)
3. (c) \(y=3e^{-0.5x}-5\)

Find the missing value in each case.

1. (a) y is directly proportional to \(x^2\), and \(y=48\) when \(x=4\). Find y when \(x=9\).
2. (b) y is inversely proportional to \(x^2\), and \(y=5\) when \(x=4\). Find the positive value of x for which \(y=20\).

The function f is defined by \(f(x)=\dfrac{3x-12}{x+4}\), \(x\neq-4\). Write down the equations of the asymptotes and the coordinates of the intercepts with the axes.

In each case say whether a linear, quadratic or exponential model is most suitable, and give a reason.

1. (a) A car loses 12% of its value every year.
2. (b) The area of a rectangle of fixed perimeter, against its width.
3. (c) A plumber charges a fixed £30 call-out fee plus £45 for every hour.

A javelin follows the path \(h(x)=-0.04x^2+1.6x+1.8\), where x is the horizontal distance in metres and h is the height in metres.

1. (a) Find the greatest height of the javelin.
2. (b) Find the horizontal distance at which the javelin reaches the ground.
3. (c) Say what the y-intercept tells you.

A metal block is taken from a furnace and left to cool. Its temperature in °C after t minutes is \(T(t)=21+179\times0.95^t\).

1. (a) Find the starting temperature of the block.
2. (b) Write down the temperature that the block approaches. Say what it represents.
3. (c) Find the time for the block to cool to 60 °C. Give your answer to 3 significant figures.

The strength S of a radio signal, in units, is inversely proportional to the square of the distance d metres from the transmitter, so \(S=\dfrac{a}{d^2}\). At 3 m the strength is 32 units.

1. (a) Find a.
2. (b) Find the strength at 8 m.
3. (c) Explain what the vertical asymptote at \(d=0\) tells you about the model.

Two sets of measurements are recorded.

1. (a) The height of a plant, in cm, after x weeks is 5, 8, 11, 14, 17 for x = 0, 1, 2, 3, 4. Choose a model, give a reason, write the model and predict the height at \(x=12\).
2. (b) The number of people who have heard a rumour after x hours is 3, 9, 27, 81 for x = 0, 1, 2, 3. Choose a model, give a reason, write the model and predict the number at \(x=6\).

The function f is defined by \(f(x)=\dfrac{ax-5}{x+3}\), \(x\neq-3\). The horizontal asymptote of the graph is \(y=2\). Find a, then find the coordinates of the intercepts with the axes.

The graph of \(y=ka^x+c\) has the horizontal asymptote \(y=-1\) and passes through \((0,\,1)\) and \((1,\,5)\). Find k, a and c, then find the value of y when \(x=4\).