Problem 33 - Algebra and Trigonometry

Question

Find the maximum or minimum value of the function. $$ f(x)=x^{2}+x+1 $$

Step-by-Step Solution

Answer

The minimum value of the function is \(\frac{3}{4}\).

Step 1: Identify the Type of Function

The given function is a quadratic function of the form \(f(x) = ax^2 + bx + c\) with coefficients \(a = 1\), \(b = 1\), and \(c = 1\). A quadratic function will have a parabolic graph.

Step 2: Determine the Direction of the Parabola

Since the coefficient \(a = 1\) is positive, the parabola opens upward. Therefore, the function has a minimum value but no maximum value.

Step 3: Find the Vertex

The vertex of a parabola, given by \(f(x) = ax^2 + bx + c\), is located at \(x = -\frac{b}{2a}\). Substituting the values of \(b\) and \(a\), we find \(x = -\frac{1}{2}\).

Step 4: Calculate the Minimum Value

Substitute \(x = -\frac{1}{2}\) back into the function to find the minimum value. Compute: \(f\left(-\frac{1}{2}\right) = \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{3}{4}\). Therefore, the minimum value of the function is \(\frac{3}{4}\).

Key Concepts

Vertex of a Parabola

Quadratic functions create a curve called a parabola when graphed. This shape always has a special point called the "vertex." The vertex can either be the highest point if the parabola opens downward or the lowest point if it opens upward. It acts like a turning point for the curve.

For a standard quadratic function expressed as \( f(x) = ax^2 + bx + c \), the x-coordinate of the vertex is calculated with the formula \( x = -\frac{b}{2a} \). This simple formula is very useful because it helps us quickly find the vertex without needing to graph the function. Once you know the x-coordinate, you plug it back into the function to find the y-coordinate and, thus, the complete vertex \((x, f(x))\).

• The vertex form of the equation helps in identifying if this vertex is a minimum or maximum point based on the direction the parabola opens.

Minimum Value of a Function

Some quadratic functions have a minimum value, especially when they open upwards. This is because the parabola forms a U shape. The lowest point in this U shape is what we call the "minimum value" of the function.

To find this minimum value, you can use the vertex of the parabola. After calculating the x-coordinate of the vertex using \( x = -\frac{b}{2a} \), substitute this back into the function \( f(x) \). This gives you the y-coordinate, which is the minimum value of the function.

• In our example \( f(x) = x^2 + x + 1 \), we calculated the minimum value to be \( \frac{3}{4} \) when \( x = -\frac{1}{2} \).

This process is quick and efficient, giving a precise result every time regarding the minimum point of the function.

Parabola Direction

The direction in which a parabola opens depends on the coefficient of the \( x^2 \) term in the quadratic equation. This is crucial because it determines whether the function has a minimum or maximum value.

• If the coefficient \( a \) is positive, the parabola opens upwards like a smiley face. This means the function will have a minimum value.
• If \( a \) is negative, the parabola opens downwards like a frown and the function will have a maximum value.

In our function \( f(x) = x^2 + x + 1 \), since \( a = 1 \), the parabola opens upwards so it has a minimum value. Understanding how the direction works lets you predict the behavior of the quadratic function before doing any calculations.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is a key starting point for analyzing these types of functions. It is generally presented as \( ax^2 + bx + c \). Knowing this form is essential because it allows us to identify the coefficients \( a \), \( b \), and \( c \), which are used in further calculations.

• \( a \): This coefficient determines the direction the parabola opens. It also affects the width of the parabola; larger absolute values make the parabola narrower.
• \( b \): This coefficient, combined with \( a \), helps locate the vertex of the parabola.
• \( c \): This constant term moves the parabola up or down on the graph without affecting its shape.

Once you have the equation in this format, finding the vertex, direction, and minimum or maximum values becomes a systematic process. It forms the foundation for understanding and working with quadratic functions.