Problem 49 - 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. $$ F(x)=3-\frac{1}{2} x^{2} $$

Step-by-Step Solution

Answer

The graph of \( F(x) = 3 - \frac{1}{2}x^2 \) has a maximum value.

Step 1: Identify the Type of Quadratic Function

The given function is a quadratic function, which means its graph will be a parabola. In general, a quadratic function can be written in the form \( f(x) = ax^2 + bx + c \).

Step 2: Determine the Leading Coefficient

The leading coefficient is the coefficient of \( x^2 \). For the given function \( F(x) = 3 - \frac{1}{2}x^2 \), the leading coefficient is \( a = -\frac{1}{2} \).

Step 3: Analyze the Sign of the Leading Coefficient

Since the leading coefficient \( a = -\frac{1}{2} \) is negative, the parabola opens downward. Parabolas that open downward always have a maximum value.

Key Concepts

Parabolas

Parabolas are the U-shaped graphs of quadratic functions. These graphs have unique mathematical properties. One of the most noticeable features is their symmetrical shape. They can either open upwards or downwards. This direction depends on the sign of their leading coefficient. Parabolas can represent many real-world phenomena, such as the trajectory of a ball or the shape of satellite dishes. Their standard mathematical form is given by the quadratic equation:\[f(x) = ax^2 + bx + c\]In this equation:

• "a" determines the direction and width of the parabola
• "b" influences the position of the vertex along the x-axis
• "c" determines the y-intercept, where the graph crosses the y-axis

Understanding how each term affects the parabola allows us to predict its shape and properties without actually plotting the graph.

Leading Coefficient

The leading coefficient, denoted as "a" in the quadratic equation \( f(x) = ax^2 + bx + c \), plays a crucial role in determining a parabola's direction. Specifically, it dictates whether the parabola opens upwards or downwards.- If "a" is positive, the parabola opens upwards. This means it has a minimum point, known as the vertex, which is the lowest point on the graph.- If "a" is negative, like in our original exercise where \( a = -\frac{1}{2} \), the parabola opens downwards. This causes a maximum point at the vertex, being the highest point on the graph.The magnitude of the leading coefficient also affects the parabola's width. The greater the absolute value of "a," the narrower the parabola appears. Conversely, smaller absolute values result in a wider parabola. Recognizing these influences helps in quickly analyzing the behavior of quadratic equations.

Quadratic Graphs

Quadratic graphs are visual representations of quadratic functions. They're most commonly associated with parabolas. Each unique quadratic function yields a distinctive parabola. However, all share some fundamental characteristics.A quadratic graph has:

• A vertex, which is either a minimum or a maximum point
• An axis of symmetry, a vertical line that runs through the vertex, dividing the parabola into two mirror-image halves
• A y-intercept, the point where the graph intersects the y-axis (found at \( c \) in the equation)

For any quadratic function, analyzing these key features makes it easier to draw or understand its graph. Mastering the concepts surrounding quadratic graphs assists in grasping more complex algebraic and calculus topics later in mathematics.