Problem 47 - 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. $$ f(x)=2 x^{2}-5 $$

Step-by-Step Solution

Answer

The graph has a minimum value.

Step 1: Understanding the function type

The given function is a quadratic function, which is generally represented as \( f(x) = ax^2 + bx + c \). In this problem, \( f(x) = 2x^2 - 5 \), which follows the form of a quadratic function.

Step 2: Identifying the parabola direction

The quadratic function \( f(x) = 2x^2 - 5 \) has a leading coefficient \( a = 2 \). Since 2 is positive, the parabola opens upwards, meaning it will have a minimum value.

Step 3: Describing the vertex

For a parabola that opens upwards, the vertex of the parabola is its minimum point. In the function \( f(x) = 2x^2 - 5 \), the parabola opens upwards, indicating that there is a minimum value.

Key Concepts

Parabola Direction

A quadratic function like \( f(x) = ax^2 + bx + c \) forms a U-shaped curve called a parabola. The direction in which this parabola opens is determined by the leading coefficient \( a \). If \( a \) is positive, the parabola opens upwards, resembling a cup. Conversely, if \( a \) is negative, the parabola opens downwards, resembling an upside-down cup.

• Upwards-opening parabola: has a minimum value.
• Downwards-opening parabola: has a maximum value.

In our exercise, the function is \( f(x) = 2x^2 -5 \) with \( a = 2 \). Since \( a \) is positive, the parabola opens upwards and therefore has a minimum point.

Vertex of a Parabola

The vertex of a parabola is an important point; it represents either the minimum or maximum value of the quadratic function. For a parabola that opens upwards, the vertex is at the lowest point on the graph. If the parabola opens downwards, the vertex is at the highest point.
Locating the vertex in a quadratic equation like \( f(x) = ax^2 + bx + c \) can be done using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \). For the function \( f(x) = 2x^2 - 5 \), because there is no \( b \) value (or \( b = 0 \)), the vertex is simply at the point \( x = 0 \). Substituting \( x = 0 \) back into the function gives us the y-coordinate, so the vertex here is \( (0, -5) \). This point is the minimum point of the parabola.

Leading Coefficient

Understanding the leading coefficient in a quadratic function is key to understanding the shape and direction of its graph. The leading coefficient, represented as \( a \) in the standard form \( f(x) = ax^2 + bx + c \), dictates the direction and width of the parabola.

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Moreover, the absolute value of \( a \) affects the width of the parabola:

• Larger absolute values of \( a \) make the parabola "steeper" or "narrower."
• Smaller absolute values make it "wider."

In our example, \( a = 2 \) suggests that the parabola is both upwards-opening and relatively narrower compared to a parabola where \( a \) is a smaller positive number. This insight helps, without any detailed calculation, to predict the parabola's general appearance.