Straight line graphs

Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is 8 cm². The area one day later is 8.8 cm².

At first, Zac uses a model of the form A = a + bt, where A cm² is the area t days after he begins measuring and a and b are constants.

Find the values of a and b that best model the initial area and the area one day later.

Calculate the value of t for which the model predicts an area of 15 cm².

Zac notices the area covered by the culture increases by 10% each day.

Explain why this model may not be suitable after the first day.

Zac decides to use a different model for A. His new model is A = Pe^kt, where P and k are constants.

Find the values of P and k that best model the initial area and the area one day later.

Calculate the value of t for which the area reaches 15 cm² according to this model.

Explain why this model may not be suitable for large values of t.

Substitute appropriate values of t₁ and t₂ to verify that t₁t₂ gives the correct value for the

y ‑coordinate of the point of intersection of the tangents at the points A and B in Fig. C1.

Show that, for the curve y = ax² + bx + c, the equation of the tangent at the point with

Hence show that for the curve with equation y = ax² + bx + c, the tangents at two points, P and Q, on the curve cross at a point which has x ‑coordinate equal to the mean of the

x ‑coordinates of points P and Q, as given in lines 11 to 14.

A student is investigating the intersection points of tangents to the curve y = 6x² - 7x + 1. She uses software to draw tangents at pairs of points with x ‑coordinates differing by 5.

Find the equation of the curve that all the intersection points lie on.

In this question you must show detailed reasoning.

This question is about the curve y = x³ - 5x² + 6x.

Find the equation of the tangent, T, to the curve at the point (0, 0).

Find the equation of the normal, N, to the curve at the point (1, 2).

Find the coordinates of the point of intersection of T and N.

Quadrilateral KLMN has vertices K (- 4, 1), L (5, - 1), M (6, 2) and N (2, 5), as shown in Fig. 6.1.

Find the coordinates of the following midpoints.

• P, the midpoint of KL

• Q, the midpoint of LM

• R, the midpoint of MN

• S, the midpoint of NK

Verify that PQRS is a parallelogram.

Show that = k(-a + b - c), where k is a constant to be determined.

Verify that AH is parallel to DE.

Verify that BC is parallel to GF.

Use your knowledge of the large data set to comment on Dr Procter’s suggestion.

Dr Procter wishes to predict the median house price in Barking and Dagenham in 2016. She uses the spreadsheet function LINEST to find the equation of the line of best fit for the given data. She obtains the equation

P = 4897 Y - 9 657 847, where P is the median house price in pounds and Y is the calendar year, for example 2015.

Use Dr Procter’s equation to predict the median house price in Barking and Dagenham in

• 2016

• 2017.

Professor Jackson uses a simpler model by using the data from 2014 and 2015 only to form a straight-line model.

Find the equation Professor Jackson uses in her model.

Use Professor Jackson’s equation to predict the median house price in Barking and Dagenham in

• 2016

• 2017.

Professor Jackson carries out some research online. She finds some information about median house prices in Barking and Dagenham, which is shown in Fig. 11.2.

Comment on how well

• Dr Procter’s model fits the data,

• Professor Jackson’s model fits the data.

Explain which, if any, of the models is likely to be more reliable for predicting median house

prices in Barking and Dagenham in 2020.

Fig. 12 shows a curve C with parametric equations x = 4t², y = 4t. The point P, with parameter t is a general point on the curve. Q is the point on the line x + 4 = 0 such that PQ is parallel to the x-axis. R is the point (4, 0).

Show algebraically that P is equidistant from Q and R.

Find a cartesian equation of C.

Show that the equation of the line in Fig. C2 is ry + hx = hr, as given in line 24.