Sketching curves: Practice Questions

Which of the following describes the shape and key feature of the graph of \(y = 2x^2 - 5x + 3\)?

Find the coordinates of the vertex (turning point) of the quadratic curve given by \(y = x^2 - 6x + 8\).

The curve \(y = x^3 - 3x^2\) is sketched. Which coordinates represent the local maximum and local minimum turning points of this curve?

What are the \(x\)-intercepts of the curve \(y = x^2 - 4x - 5\)?

Consider the reciprocal function \(y = \frac{2}{x - 3} + 1\). What are the equations of its vertical and horizontal asymptotes?

The graph shows the quadratic function \(f(x) = x^2 - 4x + 3\). Use the graph to state the coordinates of the vertex.

The graph of the function \(y = \frac{1}{x + a} + b\) has a vertical asymptote at \(x = 4\) and a horizontal asymptote at \(y = -2\). Determine the values of \(a\) and \(b\).

Consider the function \(f(x) = \frac{2x + 1}{x - 2}\).

(a) State the equation of the vertical asymptote of the graph of \(y = f(x)\).

(b) State the equation of the horizontal asymptote of the graph of \(y = f(x)\).

(c) Find the coordinates of the point where the curve intersects the \(y\)-axis.

Consider the function \(y = x^2 - 4\).

(a) Complete the table of values below for \(y = x^2 - 4\).

| \(x\) | -3 | -2 | -1 | 0 | 1 | 2 | 3 |

| \(y\) | | | | | | | |

(b) Plot these points on a grid and draw the graph of \(y = x^2 - 4\) for \(-3 \le x \le 3\).

Consider the functions \(f(x) = 2x + 1\) and \(g(x) = x^2 - 2\).

(a) Sketch the graphs of \(y = f(x)\) and \(y = g(x)\) on the same set of axes for \(-3 \le x \le 3\).

(b) Use your graph to find the approximate coordinates of the points of intersection.