- The function \(f(x) = (x - 2)(x - 4)\) is in the intercept form (Option D).
- Since the leading coefficient is positive, the parabola opens upward (Option B).
- The x-intercepts are found by setting the factors to zero, resulting in (2, 0) and (4, 0) (Option A).
- By checking other characteristics, we find that the y-intercept is (0, 8) and the vertex is (3, -1).
- The correct options are: \(\boxed{A, B, D}\).
Explanation
- Identify the Form of the Function
To analyze the quadratic function \(f(x) = (x - 2)(x - 4)\), we first identify its mathematical form. This expression is in the form \(f(x) = a(x - p)(x - q)\), where \(a = 1\), \(p = 2\), and \(q = 4\). This is known as the intercept form of a quadratic function, confirming that option D is correct.
- Determine the Direction of the Parabola
Since the leading coefficient \(a = 1\) is greater than 0, the parabola opens upward. This means option B is correct and option E is incorrect.
- Calculate the x-intercepts
To find the \(x\)-intercepts, we set \(f(x) = 0\):
\[(x - 2)(x - 4) = 0\]
This equation is satisfied when \(x - 2 = 0\) or \(x - 4 = 0\), giving us \(x = 2\) and \(x = 4\). Thus, the \(x\)-intercepts are the points \((2, 0)\) and \((4, 0)\), which matches option A.
- Calculate the y-intercept
To find the \(y\)-intercept, we evaluate the function at \(x = 0\):
\[f(0) = (0 - 2)(0 - 4) = (-2)(-4) = 8\]
The \(y\)-intercept is the point \((0, 8)\). Option C suggests the intercepts are \((0, 2)\) and \((0, 4)\), which is incorrect.
- Calculate the Vertex
The vertex \((h, k)\) of a quadratic function in intercept form can be found by taking the midpoint of the \(x\)-intercepts:
\[h = \frac{2 + 4}{2} = 3\]
To find the \(y\)-coordinate \(k\), we substitute \(h\) back into the function:
\[k = f(3) = (3 - 2)(3 - 4) = (1)(-1) = -1\]
The vertex is \((3, -1)\). Option F incorrectly states the vertex is \((2, 4)\). Consequently, the correct options are A, B, and D.
Examples
The intercept form of a quadratic function is particularly useful when designing physical structures like arched bridges or parabolic satellite dishes. By knowing the points where the arch hits the ground (the x-intercepts), engineers can easily determine the midpoint and height of the arch without needing to convert the equation into other forms, allowing for precise and efficient construction.