Problem 3 - Algebra and Trigonometry

Question

The graph of \(f(x)=2(x-3)^{2}+5\) is a parabola that opens ________, with its vertex at ( _____, ____) and \(f(3)=\) _________ is the (minimum/maximum) _________ value of \(f\)

Step-by-Step Solution

Answer

Upwards, vertex at (3, 5), minimum value is 5.

Step 1: Determine Direction of Opening

The function given is in the standard form of a parabola, \(f(x) = a(x - h)^2 + k\). Here, \(a = 2\), \(h = 3\), and \(k = 5\). Since \(a\) is positive, the parabola opens upwards.

Step 2: Identify Vertex

The vertex of the parabola is given by the point \((h, k)\). From the function, \(h = 3\) and \(k = 5\), so the vertex is \((3, 5)\).

Step 3: Evaluate f at x = 3

To find \(f(3)\), substitute \(x = 3\) into the function: \(f(3) = 2(3 - 3)^2 + 5 = 2(0)^2 + 5 = 5\).

Step 4: Determine the Minimum/Maximum Value

Since the parabola opens upwards, the vertex represents the minimum value of the function. Therefore, \(f(3) = 5\) is the minimum value.

Key Concepts

Vertex

The vertex is a fundamental part of a parabola. It represents a point at which the parabola changes direction. In the standard form of a parabola equation, which is written as \[ f(x) = a(x - h)^2 + k, \] - The values of \(h\) and \(k\) give the coordinates of the vertex. In our given exercise, the function is \[ f(x) = 2(x - 3)^2 + 5. \]This means that the vertex is at the point \((h, k) = (3, 5).\)The vertex essentially acts as a balancing point of the parabola, where the shape transitions smoothly downwards or upwards, depending on the opening. This is what makes it a crucial component for understanding the nature and position of the parabola graph. It is especially useful when determining maximum or minimum values, as those values occur precisely at the vertex.

Minimum Value

The minimum value of a parabola is the lowest point a function can reach, depending on its direction of opening. For parabolas that open upwards, the vertex itself marks this minimum value.For the given function \[ f(x) = 2(x - 3)^2 + 5, \]the vertex \((3, 5)\)provides us with crucial information:- Since the parabola is opening upwards, the \(y\)-coordinate of the vertex, \(k = 5\), is the lowest that the \(f(x)\) value can attain.Thus, at the point where \(x = 3\), the function takes on this absolute minimum value. Knowing this helps in analyzing and predicting the behavior of the quadratic function, making it easier to solve related algebraic problems or even inequations involving similar expressions.

Direction of Opening

The direction in which a parabola opens is determined by the coefficient \(a\) in its standard form \[ f(x) = a(x - h)^2 + k. \]- If \(a > 0\), the parabola opens upwards.- Conversely, if \(a < 0\), it opens downwards.In our particular problem:- We have \(a = 2\), which is greater than zero, indicating that the parabola opens upwards.Understanding the direction of opening is important because it immediately tells us whether the vertex represents a minimum or a maximum point. This property impacts how the function behaves as \(x\) increases or decreases and becomes crucial in practical applications, like determining the range of the function or inspecting its graphical representation.