Topic 2.3: Transformations of Graphs Questions

Use your GDC to find the coordinates of the minimum point of f(x) = x² − 4x + 1.

Find the y-intercept of f(x) = −3x + 8.

Use your GDC to find the x-intercepts of f(x) = x² + x − 12.

A diver's height above water (in metres) is modelled by h(t) = −5t² + 4t + 9, where t is seconds after jumping. State the diver's starting height above the water.

A model rocket is fired straight up. Its height, h metres, t seconds after launch is given by h(t) = −5t² + 30t, for 0 ≤ t ≤ 6.

(a) Find the x-intercepts of the graph of h.

(b) Find the maximum height of the rocket and the time at which it occurs.

(c) Sketch the graph of h for 0 ≤ t ≤ 6, labelling the intercepts and the maximum point.

The number of bacteria in a sample, N, after t hours is modelled by N(t) = 150 × 1.4ᵗ.

(a) Find the y-intercept of the graph of N.

(b) State what the y-intercept represents in this context.

f(x) = x² − 5x + 6.

Find the x-intercepts of g(x) = 2x² + x − 6.

A boat's speed v (in km/h) is modelled by v(t) = 18 − 0.6t, where t is hours after midnight. Find the time at which the boat stops, and interpret your answer in context.

Two lines have equations f(x) = 3x − 2 and g(x) = −x + 6. Use your GDC to find their point of intersection.

Use your GDC to find the maximum point of f(x) = −2x² + 12x − 10.

For each graph described below, write down the family of function it most likely belongs to (linear, quadratic, exponential, or sinusoidal).

(a) A curve that rises ever more steeply and gets close to, but never touches, the x-axis on the left.

(b) A smooth ∪-shaped curve with one lowest point.

(c) A wave that repeats the same up-and-down pattern every 8 units.

A market stall's daily revenue is R(x) = −0.5x² + 12x and its daily cost is C(x) = 4x + 30, where x is the number of items sold and both are measured in dollars.

Use your GDC to find the two break-even points — the values of x where revenue equals cost — and state the revenue at each.

A projectile's height (m) is modelled by h(t) = 80t − 5t², where t is seconds after launch. Suggest GDC window settings (Xmin, Xmax, Ymin, Ymax) that show the entire flight.

A company's profit (thousand $) for x months of operation is P(x) = −0.5x² + 6x − 5. Use your GDC to answer the following.

A bakery's daily profit ($) for selling x pies is P(x) = −x² + 24x − 80. Find both intercepts of P and explain what each means in the context of the bakery.

The curved cross-section of a tunnel entrance is modelled by h(x) = −x² + 10x − 16, where h is the height in metres and x is the horizontal distance in metres from a fixed point on the ground.

(a) Find the x-intercepts of the graph of h.

(b) Hence find the width of the tunnel at ground level.

Water from a fountain follows a path modelled by h(x) = −0.25x² + 3x, where h is the height in metres and x is the horizontal distance in metres from the nozzle.

(a) Use your GDC to find the x-intercepts of the graph of h.

(b) Hence write down the horizontal distance the water travels before landing.

(c) Use your GDC to find the maximum height of the water.