More about graphs of functions：練習問題

If the graph of \(y = f(x)\) is reflected across the \(y\)-axis, which of the following represents the equation of the new graph?

The graph of $$y = f(x)$$ is transformed to $$y = f(x-1) + 3$$. Which of the following describes the transformation?

The graph of \(y = f(x)\) has a local minimum at the point \((2, -2)\) and a local maximum at the point \((-1, 3)\). If \(g(x) = 1 - 2f(2x + 4)\), what is the local maximum value of the function \(g(x)\)?

The function \(f(x)\) is defined by \(f(x) = x^2 - 4x + 5\). If the graph of \(y = g(x)\) is obtained by reflecting the graph of \(y = f(x)\) in the x-axis, what is the algebraic expression for \(g(x)\)?

The graph of \(y = f(x)\) is a parabola that intersects the x-axis at \(x = -1\) and \(x = 3\). The parabola opens upwards.

For which range of values of \(x\) is \(f(x) < 0\)?

Find the maximum value of the function \( y = 5 - 2\sin x \).

The domain of the function \( y = g(x) \) is \( 0 \le x \le 12 \). Find the domain of the function \( y = g\left(\frac{x}{3}\right) \).

The function is given by \(f(x) = 2 \cos(x)\). The graph of \(y = f(x)\) is transformed to the graph of \(y = g(x)\) by the relation \(g(x) = 3f(2x - \pi) + 1\). Determine the range of \(g(x)\) and the period of the function \(g(x)\).

Consider the function \(f(x) = x^2 + 2x\). The graph of \(y = g(x)\) is obtained by reflecting the graph of \(y = f(x)\) about the x-axis and then translating the resulting graph vertically 5 units downwards.

(a) Express \(g(x)\) in the form \(ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants.
(b) Find the coordinates of the vertex of the graph of \(y = g(x)\).
(c) State the y-intercept of the graph of \(y = g(x)\).

Consider the quadratic function $$f(x) = x^2 - 8x + 12$$.

(a) Express $$f(x)$$ in the form $$(x-h)^2 + k$$ by completing the square.

(b) Hence, state the coordinates of the vertex and the axis of symmetry of the graph of $$y = f(x)$$.

(c) Find the $$x$$-intercepts and the $$y$$-intercept of the graph of $$y = f(x)$$.