Key features and models of graphs Problem set A

The function \(g(x)=-2x^2+12x-10\) 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) \(y=3\times2^x-4\)
2. (b) \(y=5\times0.5^x+2\)
3. (c) \(y=-e^{-x}+7\)

Find the missing value in each case.

1. (a) y is directly proportional to \(x^2\), and \(y=45\) when \(x=3\). Find y when \(x=7\).
2. (b) y is inversely proportional to \(x^2\), and \(y=2\) when \(x=6\). Find the positive value of x for which \(y=8\).

The function f is defined by \(f(x)=\dfrac{4x+8}{x-2}\), \(x\neq2\). 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) The number of bacteria in a dish doubles every hour.
2. (b) The height of a stone thrown upwards, against time.
3. (c) A taxi charges a fixed £4 plus £1.50 for every kilometre.

A basketball player's shot follows the path \(h(x)=-0.2x^2+1.2x+2\), where x is the horizontal distance in metres and h is the height in metres.

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

A hot drink cools in a room. Its temperature in °C after t minutes is \(T(t)=18+62\times0.9^t\).

1. (a) Find the starting temperature of the drink.
2. (b) Write down the temperature that the drink approaches. Say what it represents.
3. (c) Find the time for the drink to cool to 25 °C. Give your answer to 3 significant figures.

The intensity I of a lamp's light, in lux, is inversely proportional to the square of the distance d metres from it, so \(I=\dfrac{a}{d^2}\). At 2 m the intensity is 45 lux.

1. (a) Find a.
2. (b) Find the intensity at 5 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 depth of water in a tank, in cm, after x minutes is 3, 7, 11, 15, 19 for x = 0, 1, 2, 3, 4. Choose a model, give a reason, write the model and predict the depth at \(x=10\).
2. (b) The number of cells in a sample after x hours is 5, 10, 20, 40 for x = 0, 1, 2, 3. Choose a model, give a reason, write the model and predict the number at \(x=7\).

The function f is defined by \(f(x)=\dfrac{ax+3}{x-2}\), \(x\neq2\). The horizontal asymptote of the graph is \(y=4\). Find a, then find the coordinates of the intercepts with the axes.

The graph of \(y=ka^x+c\) has the horizontal asymptote \(y=3\) and passes through \((0,\,7)\) and \((1,\,11)\). Find k, a and c, then find the value of y when \(x=5\).