Graphs of functions：練習問題

Which of the following points lies on the graph of the function \( f(x) = x^2 - 3x \)?

Which of the following functions has a graph that passes through the point \((2, 11)\)?

In the diagram, a parabola with the equation \( y = x^2 - x - 6 \) intersects a straight line with the equation \( y = 2x - 2 \).
Find the coordinates of the two points of intersection.

Which of the following describes the graph of the function \( f(x) = -3x + 4 \)?

Find the coordinates of the minimum point (vertex) of the quadratic function \( f(x) = x^2 + 8x + 15 \).

A function is defined as \( f(x) = (x - 2)(x + 5) \). Write down the x-coordinates of the two points where the graph of this function crosses the x-axis.

Find the coordinates of the vertex of the parabola defined by the equation \( y = x^2 - 6x + 5 \).

A function is defined by \(f(x) = x^3 + px^2 + qx - 12\). The graph of \(y = f(x)\) passes through the points \((2, 0)\) and \((-1, -18)\).

(a) Use these points to find the values of \(p\) and \(q\).

(b) Using the values from part (a), find the exact coordinates of the three points where the graph intersects the \(x\)-axis.

The graph of a function \(f(x)\) is given.

(Assume a sketch is provided in an actual exam, showing a curve passing through \((-1, 0)\), \((0, 2)\), \((1, 3)\), \((2, 2)\), \((3, 0)\), \((4, -3)\). It looks like a downward-opening parabola shifted upwards).

Use the graph to estimate the following values:

(a) The value of \(f(0)\).

(b) The values of \(x\) for which \(f(x) = 0\).

The graph of a quadratic function \(y = ax^2 + bx + c\) has a vertex at \((1, -2)\) and passes through the point \((3, 6)\).

(a) Write the equation of the quadratic function in the form \(y = a(x - h)^2 + k\).

(b) Expand your equation from part (a) into the form \(y = ax^2 + bx + c\).

(c) Find the x-intercepts of the graph, correct to 2 decimal places.