Problem 3 - Precalculus Mathematics for Calculus

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

The graph opens upwards with its vertex at (3, 5) and \(f(3) = 5\) is the minimum value of \(f\).

Step 1: Identify the direction of the parabola

The function given is in the form of vertex form of a parabola: \[ f(x) = a(x-h)^2 + k \]where \(a\) determines the direction the parabola opens. If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. Here, \(a = 2\), so the parabola opens upwards.

Step 2: Find the vertex of the parabola

The vertex form of a parabola is \[ f(x) = a(x-h)^2 + k \]In the equation provided, \[ f(x) = 2(x-3)^2 + 5 \]\(h\) is 3, and \(k\) is 5, so the vertex of the parabola is at \((3, 5)\).

Step 3: Calculate the function value at x = 3

Since the vertex x-value is 3, we can substitute \(x = 3\) into the equation to find \(f(3)\): \[ f(3) = 2(3-3)^2 + 5 = 2(0)^2 + 5 = 5 \]Therefore, \(f(3)\) is 5.

Step 4: Determine if the value is a minimum or maximum

Since the parabola opens upwards, the value at the vertex \((3, 5)\) is a minimum value, as it is the lowest point on the graph of the function.

Key Concepts

Direction of a Parabola

Understanding the direction in which a parabola opens is a key concept in quadratic functions. For the vertex form of a parabola, which is expressed as \( f(x) = a(x-h)^2 + k \), the value of \( a \) plays a crucial role in determining the direction.

• If \( a > 0 \), the parabola opens upwards. This means that the parabola forms a "U" shape, and the arms of the curve extend towards positive infinity.
• If \( a < 0 \), the parabola opens downwards. This causes the parabola to form an upside-down "U" or an "n" shape, where the arms extend towards negative infinity.

In the example function \( f(x) = 2(x-3)^2 + 5 \), \( a \) is 2, which is greater than zero. Therefore, the parabola opens upwards, indicating that the curve has a lowest point on the graph.

Minimum Value of a Function

The minimum or maximum value of a quadratic function depends on the direction in which the parabola opens. This value is found at the vertex of the parabola.

• If the parabola opens upwards (\( a > 0 \)), the function has a minimum value. The vertex represents the lowest point on the graph.
• If the parabola opens downwards (\( a < 0 \)), then the function exhibits a maximum value, with the vertex being the highest point.

Using our function \( f(x) = 2(x-3)^2 + 5 \), since it opens upwards because \( a = 2 \), the function's vertex \(3, 5\)\ is a minimum. The minimum value of the function is \( f(3) = 5 \), as calculated by plugging \( x = 3 \) into the equation.

Vertex of a Parabola

The vertex of a parabola is a critical point that provides valuable information about the graph of a quadratic function. It is given in the expression \( f(x) = a(x-h)^2 + k \) as the point \((h, k)\). The vertex serves as either the highest or lowest point on the graph.

• The \( h \) value represents the x-coordinate of the vertex. This is the axis of symmetry for the parabola, meaning the curve is mirrored evenly across this vertical line.
• The \( k \) value signifies the y-coordinate, detailing how far up or down the vertex sits from the x-axis.

For the function \( f(x) = 2(x-3)^2 + 5 \), the vertex is at \((3, 5)\). This is because \( h = 3 \) and \( k = 5 \). Thus, this specific point determines the parabola's direction and the minimum or maximum value of the function.