Problem 50 - Algebra A Combined Function

Question

Without calculating, tell whether each graph has a minimum value or a maximum value. See the Concept Check in the section. $$ G(x)=3-\frac{1}{2} x+0.8 x^{2} $$

Step-by-Step Solution

Answer

The graph has a minimum value.

Step 1: Analyze the function type

The function given is a quadratic function as it is of the form \( G(x) = ax^2 + bx + c \). In this case, \( a = 0.8 \), \( b = -\frac{1}{2} \), and \( c = 3 \). Quadratic functions represent parabolas.

Step 2: Determine the direction of the parabola

In quadratic functions, the parabola opens upwards if the coefficient \( a \) is positive, and downwards if \( a \) is negative. Since \( a = 0.8 \) and it is positive, the parabola opens upwards.

Step 3: Identify the presence of a minimum or maximum value

For an upwards-opening parabola, the lowest point of the parabola is called the minimum value. Therefore, this particular quadratic function has a minimum value.

Key Concepts

Understanding Parabolas

When discussing quadratic functions, one of the most significant aspects to understand is the shape of the graph, known as a parabola. A parabola is a symmetrical, curved shape that can open either upwards or downwards. The direction of the opening depends on the quadratic function's leading coefficient: the coefficient of the squared term (i.e., the term containing \(x^2\)). In any quadratic function of the form \( ax^2 + bx + c \), this is the \(a\) value. For instance, in our example, \(a=0.8\). Because this value is positive, we know the parabola opens upwards.

• If \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, the parabola opens downwards.

Understanding the direction of the parabola is crucial in determining whether a graph has a minimum or maximum value.

Determining the Minimum Value

A minimum value of a parabola occurs at its lowest point. This concept is particularly important in fields like physics and economics where optimization is key. When a quadratic function opens upwards, as seen with our example where \(a=0.8\), it indicates the parabola has a minimum value. This is because the lowest point is reached at the vertex.
The vertex of an upwards-opening parabola can be thought of as the 'bottom' of the curve. Mathematically, the vertex is at the point \( x = -\frac{b}{2a} \), and the calculation of the function at this point will give you the minimum value.

• Minimum value indicates the least point on an upwards-opening parabola.
• You can find this value using the vertex formula to easily determine the vertex position.

Importance of Coefficients

Coefficients play a vital role in understanding the behavior of quadratic functions. These numbers, which multiply the variables, essentially shape the parabola and determine several of its features. In the equation \( ax^2 + bx + c \), the coefficients \(a\), \(b\), and \(c\) can individually affect the graph:

• \(a\): Determines the direction and the 'width' or 'narrowness' of the parabola. A larger absolute value of \(a\) results in a narrower parabola.
• \(b\): Affects the position of the vertex along the horizontal axis. It influences the symmetrically balanced aspect of the graph.
• \(c\): Represents the y-intercept, the point where the parabola crosses the y-axis. This is the value of the quadratic function when \(x=0\).

Therefore, knowing the coefficients isn't just about filling in numbers. It's about predicting and understanding the shape and position of the parabola on a graph.