Problem 35 - Algebra and Trigonometry

Question

\(29-38=\) Find the maximum or minimum value of the function. $$ h(x)=\frac{1}{2} x^{2}+2 x-6 $$

Step-by-Step Solution

Answer

The minimum value of the function is -8 at \( x = -2 \).

Step 1: Identify the Vertex Form of a Quadratic Function

The function is given as a quadratic function: \( h(x) = \frac{1}{2}x^2 + 2x - 6 \). A quadratic function in the form \( ax^2 + bx + c \) has a standard method to find its maximum or minimum by identifying the vertex of the parabola, which is the point \((x, h(x))\) where the function attains its extremum. The vertex formula for \(x\) is given by \( x = -\frac{b}{2a} \).

Step 2: Calculate the Vertex x-coordinate

Here, \( a = \frac{1}{2} \), \( b = 2 \), and \( c = -6 \). Plug \( a \) and \( b \) into the vertex formula to find the \( x \)-coordinate of the vertex:\[ x = -\frac{b}{2a} = -\frac{2}{2 \times \frac{1}{2}} = -2 \].

Step 3: Evaluate the Function at the Vertex

Substitute \( x = -2 \) back into the original function to find the \( y \)-coordinate of the vertex:\[ h(-2) = \frac{1}{2}(-2)^2 + 2(-2) - 6 \].Simplify this:\[ h(-2) = \frac{1}{2} \times 4 - 4 - 6 = 2 - 4 - 6 = -8 \].

Step 4: Determine the Nature of the Vertex

Since the coefficient of \( x^2 \), \( \frac{1}{2} \), is positive, the parabola opens upwards. Therefore, the vertex \((-2, -8)\) is a minimum point.

Key Concepts

Vertex Form

To better understand how to find the maximum or minimum value of a quadratic function, it's essential to discuss the vertex form. The vertex form of a quadratic function is given by \( f(x) = a(x - h)^2 + k \). Here, \( (h, k) \) represents the vertex of the parabola, which is the highest or lowest point on the graph, depending on the direction it opens. The coefficient "\( a \)" determines whether the parabola opens upwards (\( a > 0 \)) or downwards (\( a < 0 \)).

• "\( h \)" is the x-coordinate of the vertex.
• "\( k \)" is the y-coordinate of the vertex.

This form is particularly useful as it makes it easy to identify the vertex directly and analyze the properties of a quadratic function quickly. Converting a quadratic function from the standard form \( ax^2 + bx + c \) into the vertex form involves completing the square, a technique that rearranges terms to reveal the vertex.

Parabola

The graph of a quadratic function is known as a parabola. Understanding a parabola and its orientation is crucial when analyzing quadratic functions. A parabola is a symmetric curve that forms a "U" shape if it opens upwards, or an upside-down "U" if it opens downwards. The orientation is determined by the sign of the leading coefficient "\( a \)" in the quadratic equation. If \( a > 0 \), the parabola opens upwards, indicating a minimum point at the vertex. If \( a < 0 \), the parabola opens downwards, showing a maximum point at the vertex.

• The axis of symmetry is a vertical line through the vertex, given by \( x = h \).
• Every point on the parabola is equidistant from the focus and the directrix, special features related to the parabola's geometric definition.

The symmetry of the parabola is a powerful feature that helps in locating the vertex and analyzing the function's behavior.

Maximum or Minimum Value of a Function

Determining whether a quadratic function has a maximum or minimum value is pivotal in many mathematical applications. This depends on the orientation of the parabola associated with the quadratic function.

• If the parabola opens upwards (\( a > 0 \)), the vertex represents the minimum value of the function.
• If the parabola opens downwards (\( a < 0 \)), the vertex signifies the maximum value of the function.

The coordinates of the vertex \((h, k)\) in the vertex form \( f(x) = a(x - h)^2 + k \), directly provide this extremum value. In the given exercise, the function \( h(x) = \frac{1}{2}x^2 + 2x - 6 \) has a minimum value at the vertex, since the parabola opens upwards (with \( a = \frac{1}{2} \)). By calculating the vertex \((-2, -8)\), we find that the minimum value of the function is \(-8\).
Understanding these concepts is essential, as it not only allows the determination of extremum values but also provides insights into the function's overall behavior.