Graphs of functions：練習問題

A linear function is given by \(f(x) = -2x + 5\). At which point does the graph of this function cross the \(y\)-axis?

Consider the quadratic function \(f(x) = x^2 - 4x + 3\). Determine the coordinates of the vertex of the graph of \(y = f(x)\).

A function is defined by \(f(x) = k \cdot a^x\). The graph passes through the points \((0, 5)\) and \((2, 20)\). If \(a > 0\), find the value of \(f(3)\).

Which of the following functions represents a reciprocal graph with a vertical asymptote at \(x = 0\)?

An exponential function is defined as \(f(x) = 2^x - 4\). State the equation of the horizontal asymptote of the graph.

Identify which of the following equations represents a linear function and which represents a quadratic function:
(i) \(y = 5x - 2\)
(ii) \(y = 2x^2 + 3x - 1\)

A function is given by \(f(x) = x^2 - 4x + 3\). By considering the shape of the graph and its intercepts, determine the coordinates of the points where the graph crosses the \(x\)-axis.

The graph of the exponential function \(f(x) = k \cdot a^x\) passes through the points \((0, 3)\) and \((2, 12)\). Given that \(a > 0\), determine the values of the constants \(k\) and \(a\).

A function is given by \(f(x) = x^2 - 4x - 5\).

(a) State the coordinates of the \(y\)-intercept.
(b) Factorise the expression for \(f(x)\) and state the \(x\)-intercepts of the graph.
(c) Determine the coordinates of the vertex of the parabola.

The graph of the function \(f(x) = \frac{a}{x} + b\) passes through the points \((2, 5)\) and \((-1, -4)\).

(a) Find the values of the constants \(a\) and \(b\).
(b) Write down the equations of the horizontal and vertical asymptotes of the graph of \(y = f(x)\).
(c) Sketch the graph of the function, showing clearly the asymptotes and any intercepts with the axes.