Functions and graphs：練習問題

For the quadratic function \(f(x) = -3(x - 2)^2 + 5\), which of the following statements is correct?

Find the minimum value of the quadratic function $$f(x) = x^2 - 6x + 11$$.

The graph of a function \(y = f(x)\) is transformed to become the graph of \(y = g(x)\) by the following sequence of transformations, applied in the given order:

1. Horizontal stretch by a factor of 2.
2. Reflection about the y-axis.
3. Vertical translation 5 units upwards.
4. Horizontal translation 1 unit to the right.

If the transformed graph \(y = g(x)\) passes through the point \((-3, 14))\) and has a horizontal asymptote at \(y = 5\), which of the following could be the equation of \(y = f(x))\)?

Find the coordinates of the vertex of the graph of the quadratic function $$f(x) = 2(x + 3)^2 - 5$$.

The graph of a function \(y = f(x)\) has a local maximum at \((-1, 8)\). The graph is transformed to obtain the graph of \(y = g(x)\) by the following sequence of transformations applied in the given order:

1. Horizontal stretch by a factor of 2.
2. Reflection about the x-axis.
3. Translation 3 units to the right.
4. Translation 1 unit downwards.

Find the coordinates of the point on the graph of \(y = g(x)\) that corresponds to the local maximum of \(y = f(x)\).

The graph of a quadratic function, y = f(x), has its vertex at the origin (0, 0). If the graph of y = f(x) is transformed to the graph of y = g(x) where g(x) = 3f(x - 4) + 5, determine the coordinates of the vertex of the graph of y = g(x).

Given the function \(f(x) = |x|\). The graph of \(y = g(x)\) is obtained by reflecting the graph of \(y = f(x)\) about the x-axis, then translating it 3 units to the right, vertically stretching it by a factor of 2, and finally translating it 5 units upwards. Find the range of \(g(x)\) for the domain \(x \in [-1, 5]\).

Find the maximum value of the quadratic function $$f(x) = -x^2 + 4x - 1$$.

Let \(f(x) = 2x^2 + 8x + 3\) be a quadratic function.

(a) Express \(f(x)\) in the form \(a(x+h)^2 + k\), where \(a\), \(h\), and \(k\) are constants. Hence, state the coordinates of the vertex of the graph of \(y = f(x)\).

(b) The graph of \(y = f(x)\) is transformed by reflecting it in the \(x\)-axis, and then translating the resulting graph 5 units upwards to obtain the graph of \(y = g(x)\).

(i) Write down the algebraic expression for \(g(x)\).

(ii) Find the range of the function \(g(x)\).

Let \(f(x)\) be a quadratic function defined by \(f(x) = x^2 - 6x + 5\).

(a) Find the coordinates of the vertex and the range of \(f(x)\).

(b) The graph of \(y = f(x)\) is transformed to obtain the graph of \(y = g(x)\) by the following sequence of transformations:
(i) Horizontal translation of 4 units to the left.
(ii) Vertical reflection about the \(x\)-axis.
(iii) Vertical stretch by a factor of 2.
(iv) Vertical translation of 1 unit upwards.
Find the algebraic expression for \(g(x)\) in the form \(ax^2 + bx + c\).

(c) Using the function \(g(x)\) found in (b), solve the inequality \(g(x) > 2(x-1)^2 - 5\).