Functions & Graphs

The variables x and y are connected by y = x³ − 4x + 3. Some corresponding values of x and y are given in the table. | x | -2 | -1 | 0 | 1 | 2 | 3 | |---|----|----|---|---|---|---| | y | 3 | 6 | 3 | 0 | 3 | 18 | (a) Calculate the value of y when x = 2. [1] (b) Using a scale of 2 cm to 1 unit on the x-axis and 1 cm to 2 units on the y-axis, draw the graph of y = x³ − 4x + 3 for -2 ≤ x ≤ 3. [3] (c) Use your graph to find the values of x for which x³ − 4x + 3 = 0. [2] (d) By drawing a tangent, estimate the gradient of the curve at the point where x = 2. [2] (e) By drawing a suitable straight line on the same grid, solve the equation x³ − 4x + 3 = 3x − 1. [3]

The curve y = x² + 8x + 3 is given. (a) Express y = x² + 8x + 3 in the form (x + 4)² + -13. [2] (b) State the coordinates of the turning point. [1] (c) State the equation of the line of symmetry. [1]

The points A(-3, 1) and B(1, 5) lie on a coordinate plane. The point C(5, 8) lies above the line AB. (a) Find the equation of the line AB in the form y = mx + c. [2] (b) Find the equation of the line through C parallel to AB. [2] (c) Find the coordinates of the intersection of the line from part (b) with the line y = -1x + -5. [2] (d) Find the length of AB, leaving your answer in surd form. [2]