Problem 13 - Algebra for College Students

Question

Determine whether the graph of each quadratic function opens upward or downward. $$f(x)=x^{2}+5$$

Step-by-Step Solution

Answer

The graph opens upward because \( a = 1 \) is positive.

Step 1: Identify the general form of a quadratic function

A quadratic function is generally written in the form \( f(x) = ax^2 + bx + c \).

Step 2: Recognize the given quadratic function

The given function is \( f(x) = x^2 + 5 \). Comparing this with the general form, it is seen that \( a = 1, b = 0, \text{and} c = 5 \).

Step 3: Determine the direction the parabola opens

The value of \( a \) determines the direction the graph of the quadratic function opens. If \( a > 0 \), the parabola opens upward. If \( a < 0 \), the parabola opens downward. In this case, \( a = 1 \).

Step 4: Conclude the direction

Since \( a = 1 \) is positive, the graph of the quadratic function \( f(x) = x^2 + 5 \) opens upward.

Key Concepts

Parabola Direction

Understanding whether a parabola opens upward or downward is essential when dealing with quadratic functions.
The general form of a quadratic function is given by:
 \( f(x) = ax^2 + bx + c \)
 The key to determining the direction of the parabola lies in the coefficient \( a \) (the term in front of \( x^2 \)). This coefficient tells us:

 - If \( a > 0 \), the parabola opens upward.
 - If \( a < 0 \), the parabola opens downward.
 

 For example, let’s consider the given quadratic function: \( f(x) = x^2 + 5 \). Here, the coefficient \( a \) is 1, which is positive.
Since \( a = 1 \) is a positive number, the graph of the function \( f(x) = x^2 + 5 \) opens upward.

Knowing this helps to predict the general shape of the graph without even plotting it.

Quadratic Formula

The quadratic formula is a crucial tool for solving quadratic equations. It is derived from the general form of a quadratic function:
 \( ax^2 + bx + c = 0 \).
 The formula helps to find the roots (or solutions) of the quadratic equation and is expressed as:
 \[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]

 Here’s a brief breakdown:
- The term \( b^2 - 4ac \) is called the discriminant. It determines the nature of the roots.
- If the discriminant is positive, there are two distinct real roots.
- If the discriminant is zero, there is exactly one real root.
- If the discriminant is negative, there are no real roots; instead, there are two complex roots.

Understanding this formula is essential for solving any quadratic equation effectively. While it does not directly tell the direction of a parabola, knowing how to find the roots adds more context and helpful information regarding the behavior of the quadratic function on the graph.

Graph Analysis

Analyzing the graph of a quadratic function provides a visual understanding of its properties and behavior. Here's what you need to focus on:

 
• Vertex: The highest or lowest point on the graph, depending on the direction the parabola opens. For \( f(x) = ax^2 + bx + c \), the vertex can be found using the formula \( (-\frac{b}{2a}, f(-\frac{b}{2a})) \).
   
 
• Axis of Symmetry: A vertical line that passes through the vertex and divides the parabola into two symmetrical halves. For the function \( f(x) = ax^2 + bx + c \), it is given by \( x = -\frac{b}{2a} \).
   
• Y-intercept: The point where the graph intersects the y-axis, given by the value of \( c \). This is always at the point \( (0, c) \).
   
• X-intercepts: The points where the graph intersects the x-axis, which are the roots of the quadratic equation \( ax^2 + bx + c = 0 \). These can be found using the quadratic formula.
   

 For the given function \( f(x) = x^2 + 5 \):
- The vertex is at (0, 5), since there is no linear term (\( b = 0 \)).
- The axis of symmetry is the vertical line \( x = 0 \).
- The y-intercept is at (0, 5).
- Since the value of \( c \) is greater than zero and \( a > 0 \), the parabola does not intersect the x-axis, meaning it has no real x-intercepts.

Analyzing these components helps to sketch the graph accurately and understand the function’s behavior in detail.