# Question 4.10b: Plot the graphs of the following: (a) y = 0.5x³ − 5x² + 8.5x......

Plot the graphs of the following:
(a) y = 0.5x³ − 5x² + 8.5x + 27         (b) y=−0.5x³ − 5x² + 8.5x + 27.
From the graphs estimate:
(i) The roots.
(ii) The turning points. Compare the graphs of (a) and (b).

Step-by-Step
The 'Blue Check Mark' means that this solution was answered by an expert.

First set up a table of values from which the graphs may be plotted. (See Table 4.8 and Figures 4.12a and 4.12b.) The roots and the turning points are summarised as follows:

(a) has only one root, x = –1.56 approximately

(a) has two turning points at x = 1.0, y = 31 and at x = 5.5, y = 5.6

(b) has three roots, at x = –11, x = –1.7 and x = 2.8 approximately

(a) has two turning points at x = –7, y = –107 and at x = 0.8, y = 30

Note: From an economic perspective, only the first quadrant (x > 0, y > 0) in Figures 4.12a and 4.12b is of interest. However, for a better understanding of mathematics, it is necessary to analyse the characteristics of the graphs in the other quadrants.

 Table 4.8 Points for plotting graphs in Worked Example 4.10b x Graph (a) Graph (b) -12 -1659 69 -8 -617 -105 -4 -119 -55 0 27 27 4 13 -51 8 31 -481 2 237 -1455

Question: 4.20

## Use the log rules to simplify the following expressions to a single term, if possible:(a) logb(25) + logb(70) – logb(55) (b) 4 logx (7) – 3 logx (0.85) + logx (10) (c) 12 logb(12) + 3 logx (8.25) – 2 logb(5) Then evaluate each expression if b = 3, x = e. ...

(a) log_b (25) + log_b (70)  –  log_b (55)[...
Question: 4.2

## Solve each of the following equations, classifying the type of solutions (roots) found as real and different, real and repeated, or complex: (a) x² + 6x + 5 =0 (b)x² + 6x + 9 =0 (c)x² + 6x + 10 = 0 ...

Use equation (4.2), the ‘minus b’ formula, to solv...
Question: 4.9

## The demand function for a good is given as Q = 65 – 5P. Fixed costs are £30 and each unit produced costs an additional £2. (a) Write down the equations for total revenue and total costs in terms of Q. (b) Find the break-even point(s) algebraically. (c) Graph total revenue and total costs on the ...

(a) TR=P×Q. Therefore, if P is written in terms of...
Question: 4.6

## (a) Sketch the graph of y = 2x² – 7x – 9 for values of x from –2 to 6. (b) Calculate the points of intersection with the axes; hence, find the roots and mark the roots of 2x² − 7x − 9 on the diagram. (c) Estimate the coordinates of the turning point from the graph. (d) Measure the difference ...

(a) Calculate the y-values given x, ranging from x...
Question: 4.3

## (a) Plot the graphs of y = x² and y=−x² on the same diagram. (b) Comment on the general properties of the graphs. What is the effect on the graph of multiplying the right-hand side of each equation by (–1)? ...

(a) Calculate a table of (x, y) points as in Table...
Question: 4.1

Question: 4.23

## Sketch the functions (a) y = 1/x − 0.23 (b) y = 1/x + 3 ...

(a) The function given in (a) is y = 1/x translate...
Question: 4.19

## Graph log(x) and ln(x) on the same diagram for −1 < x < 2. From the graphs summarise the properties of the log functions. How does the size of the base affect the shape of the graph? ...

Calculate a table of values for each function, usi...
Question: 4.13

## Simplify each of the following expressions, giving your answers with positive indices only:(a) 25/2³ 2^−4 (b) Q Q^1.5 P (c) √3^5 /3^−4 3^4 (d) 15 e^α e^β / e² (e) ( 3 L^0.5 / L^-2 )² (f) (3 e^0.5 / e^-2)² (g) p^5 q − p^4 q² / p^5 − p^4 q (h) 4e^α+2β − 2e^α / 2e^α ...

(a) ${\frac{2^{5}}{2^{3}2^{-4}}}$ = [...