Problem 17 - College Algebra

Question

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry. $$ f(x)=4 x^{2}+x-1 $$

Step-by-Step Solution

Answer

Minimum value is \(-\frac{17}{16}\); axis of symmetry is \(x = -\frac{1}{8}\).

Step 1: Identify the Form of the Quadratic Function

The given function is a quadratic function in the standard form \(f(x) = ax^2 + bx + c\), where \(a = 4\), \(b = 1\), and \(c = -1\).

Step 2: Determine the Type of Extremum

Since the coefficient \(a = 4\) is positive, the parabola opens upwards. This means there is a minimum value for the quadratic function.

Step 3: Find the Axis of Symmetry

The axis of symmetry for a quadratic function \(f(x) = ax^2 + bx + c\) is given by the formula \(x = -\frac{b}{2a}\). Substitute \(b = 1\) and \(a = 4\) to find the axis of symmetry:\[x = -\frac{1}{2(4)} = -\frac{1}{8}\].

Step 4: Calculate the Minimum Value

To find the minimum value of \(f(x)\), substitute \(x = -\frac{1}{8}\) back into the function:\[f\left(-\frac{1}{8}\right) = 4\left(-\frac{1}{8}\right)^2 + \left(-\frac{1}{8}\right) - 1\]Calculate each part:\[ = 4\left(\frac{1}{64}\right) - \frac{1}{8} - 1\]\[ = \frac{4}{64} - \frac{1}{8} - 1\]\[ = \frac{1}{16} - \frac{2}{16} - \frac{16}{16}\]\[ = -\frac{17}{16}\].Thus, the minimum value of the function is \(-\frac{17}{16}\).

Key Concepts

Understanding the Axis of Symmetry

In a quadratic function, the axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex of the parabola, serving as the "balancing point." For any quadratic function written in the form \( f(x) = ax^2 + bx + c \), you can find the axis of symmetry using the formula:

• \( x = -\frac{b}{2a} \)

This simple formula gives us the x-coordinate of the vertex, helping us understand where the parabola is centered. In our specific example, with \( a = 4 \) and \( b = 1 \), substituting these into the formula gives:

• \( x = -\frac{1}{2(4)} = -\frac{1}{8} \)

So, \( x = -\frac{1}{8} \) is the line of symmetry for the parabola described by the function. This means our parabola is perfectly symmetrical around this vertical line. Finding the axis of symmetry is crucial when graphing because it provides a starting point to understand the parabola's shape.

Finding the Minimum Value of a Quadratic Function

A quadratic function can have either a minimum or maximum value, which occurs at the vertex of the parabola. The direction the parabola opens (upwards or downwards) depends on the sign of the coefficient \( a \) in the standard form equation \( f(x) = ax^2 + bx + c \). If \( a \) is positive, like in our example where \( a = 4 \), the parabola opens upwards and has a minimum value at its vertex.To find the minimum value, we first find the x-coordinate of the vertex using the axis of symmetry formula \( x = -\frac{b}{2a} \), which we determined as \( x = -\frac{1}{8} \). Next, substitute this back into the original function to find the minimum value.For \( f(x) = 4x^2 + x - 1 \):

• Substitute \( x = -\frac{1}{8} \)
• \( f\left(-\frac{1}{8}\right) = 4\left(-\frac{1}{8}\right)^2 + \left(-\frac{1}{8}\right) - 1 \)
• Simplify to find \( f\left(-\frac{1}{8}\right) = -\frac{17}{16} \)

Thus, the minimum value of the quadratic function is \(-\frac{17}{16}\). This tells us the lowest point the parabola reaches on the y-axis.

Quadratic Functions and the Standard Form

Quadratic functions are equations that can be expressed in the form \( f(x) = ax^2 + bx + c \). This is known as the standard form. In this expression:

• \( a \) is the coefficient of \( x^2 \) and determines the direction of the parabola (whether it opens upwards or downwards)
• \( b \) is the coefficient of \( x \) and influences the position of the vertex along the x-axis
• \( c \) is the constant term, affecting where the parabola crosses the y-axis

Understanding the standard form is important for:

• Quickly identifying the parabola's orientation based on the sign of \( a \)
• Finding the vertex using the axis of symmetry
• Graphing the function accurately on a coordinate plane

The standard form is a foundational concept in algebra as it provides a straightforward method to analyze and graph quadratic functions effectively. By knowing the values of \( a \), \( b \), and \( c \), we can predict the shape and position of the parabola, which is key in solving many quadratic-related problems.