With regards to question J:
The variables x and y are connected by the equation y = x2 - x - 5. Some corresponding values of x and y are given in the table below.
x -4 -3 -2 -1 0 1 2 3 4 5
y 15 7 a -3 -5 b -3 1 7 15
(a) Calculate the values of a and b
(b) Use 2 cm to represent 1 unit on the x-axis and 1 cm to represent
1 unit on the y-axis, draw the graph of y = x2 - x - 5 for
- 4 £ x £ 5.
(c) Use the graph to find
(i) the value of y when x = - 3.5
(ii) the values of x when y = - 2
(iii) the minimum value of y
(iv) the values of x when y = 0
(v) the value of y when x = 0
(vi) the range of values of y when -2 < x < 2
(d) By drawing a suitable straight line to intersect the curve graph,
find the solutions to the equation
(i) x2 - x - 5 = 0
(ii) x2 - x - 5 = 2x + 1
(iii) x2 - x = 8
(e) Use your graph to solve the inequality x2 - x - 5 > 2x + 1
(f) By drawing a tangent, find the gradient of the curve y = x2 - x - 5
at the point (2, -3)
(g) By drawing a suitable tangent to the curve, find the co-ordinates of
the point A at which the gradient of the tangent at A is 1
(h) From the graph, find the values of x for which the gradient is positive.
(i) Describe fully the symmetry of the graph.
(j) Find the cubic equation which is satisfied by the x values of the intersecting point of the curve x2 - x - 5 and the curve 1/x
How do one go about solving it?
thks!
To solve question J, we will go through each part individually.
(a) To calculate the values of a and b, we can check the table provided. In the table, we can see that when x = -2, y is not given, so we need to determine the value of a. Similarly, when x = 1, y is not given, so we need to determine the value of b.
Looking at the equation y = x^2 - x - 5, we can substitute the values of x = -2 and x = 1 into the equation and solve for y:
For x = -2: y = (-2)^2 - (-2) - 5 = 4 + 2 - 5 = 1
So, a = 1.
For x = 1: y = (1)^2 - (1) - 5 = 1 - 1 - 5 = -5
So, b = -5.
Therefore, the values of a and b are a = 1 and b = -5.
(b) To draw the graph of y = x^2 - x - 5 for -4 ≤ x ≤ 5, we can plot the points given in the table on a graph with x on the x-axis and y on the y-axis. Each unit on the x-axis represents 2cm, and each unit on the y-axis represents 1cm.
Plot the points from the table (-4, 15), (-3, 7), (-2, 1), (-1, -3), (0, -5), (1, -5), (2, -3), (3, 1), (4, 7), and (5, 15) on the graph. Then, connect the points with a smooth curve.
(c) We will use the graph to find the following:
(i) To find the value of y when x = -3.5, we locate the point on the graph where x = -3.5 and then read the corresponding y-value. Estimate the value by looking at the graph.
(ii) To find the values of x when y = -2, we locate the point on the graph where y = -2 and then read the corresponding x-values. Estimate the values by looking at the graph.
(iii) To find the minimum value of y, we look for the lowest point on the graph.
(iv) To find the values of x when y = 0 (x-intercepts), we locate the points on the graph where the curve intersects the x-axis and read the corresponding x-values.
(v) To find the value of y when x = 0, we locate the point on the graph where x = 0 and then read the corresponding y-value.
(vi) To find the range of values of y when -2 < x < 2, we locate the highest and lowest points on the graph within the given range and read the corresponding y-values.
(d) To find the solutions to the given equations by drawing a suitable straight line to intersect the curve graph, we plot the graph of y = x^2 - x - 5 and then draw lines that represent the given equations. The points where the lines intersect the curve will give us the solutions to the equations.
(i) x^2 - x - 5 = 0: Draw a line y = 0 on the graph and find the x-values where the line intersects the curve. These x-values will be the solutions to the equation.
(ii) x^2 - x - 5 = 2x + 1: Rearrange the equation to x^2 - 3x - 6 = 0. Draw a line y = 2x + 1 on the graph and find the x-values where the line intersects the curve. These x-values represent the solutions to the equation.
(iii) x^2 - x = 8: Rearrange the equation to x^2 - x - 8 = 0. Draw a line y = 0 on the graph and find the x-values where the line intersects the curve. These x-values will be the solutions to the equation.
(e) To solve the inequality x^2 - x - 5 > 2x + 1 using the graph, we shade the area on the graph that represents the region where the curve is above the line y = 2x + 1. The shaded area will give us the solutions to the inequality.
(f) To find the gradient (slope) of the curve y = x^2 - x - 5 at the point (2, -3), draw a tangent line to the curve at that point and calculate the gradient of the tangent line.
(g) To find the coordinates of the point A where the gradient of the tangent at A is 1, draw a suitable tangent to the curve and find the point(s) where the gradient is 1.
(h) To find the values of x for which the gradient is positive, locate the points on the graph where the curve has a positive gradient, which means the curve is increasing in that region.
(i) To describe fully the symmetry of the graph, determine if the graph is symmetric about the y-axis, x-axis, or origin. Note any patterns or similarities in the graph.
(j) To find the cubic equation satisfied by the x-values of the intersecting point of the curve y = x^2 - x - 5 and the curve 1/x, we need to find the points where the two curves intersect. This can be done by solving the simultaneous equations of the two equations.
Once we have the points of intersection, we can write the equation of the cubic equation using those x-values.
Remember to refer to the graph that you have drawn for various parts of the question to provide accurate answers.