Problem 12 - Beginning and Intermediate Algebra

Question

Find the vertex of each parabola. $$ f(x)=x^{2}-x+5 $$

Step-by-Step Solution

Answer

The vertex is \( (\frac{1}{2}, \frac{19}{4}) \).

Step 1: Identify the coefficients

The general form of a quadratic function is given by \(f(x) = ax^2 + bx + c\). Here, the coefficients are: \ a = 1, b = -1, c = 5.

Step 2: Use the vertex formula

The x-coordinate of the vertex of the parabola can be found using the formula \( x = -\frac{b}{2a}\). Substitute the values of a and b: \ x = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \

Step 3: Find the y-coordinate

Substitute \(x = \frac{1}{2}\) back into the function to find the y-coordinate: \ f(\frac{1}{2}) = (\frac{1}{2})^2 - \frac{1}{2} + 5 = \frac{1}{4} - \frac{1}{2} + 5 = \frac{1}{4} - \frac{2}{4} + 5 = - \frac{1}{4} + 5 = \frac{19}{4} \

Step 4: State the vertex

The vertex of the parabola is at the point \( (\frac{1}{2}, \frac{19}{4}) \).

Key Concepts

quadratic function

A quadratic function is a type of polynomial function that is represented by the form: \( f(x) = ax^2 + bx + c \). Here:

• represents the coefficient of the term.


• is the coefficient of the term.


• is the constant term.

Quadratic functions create parabola-shaped graphs. These parabolas can open upwards or downwards, depending on the sign of . If is positive, the parabola opens upwards. If is negative, it opens downwards.Understanding the structure of a quadratic function is essential for finding its vertex and other properties.

vertex formula

The vertex of a parabola is the point where it changes direction. For quadratic functions in the form \( f(x) = ax^2 + bx + c \), you can find the vertex using the vertex formula: \( x = -\frac{b}{2a} \). This formula calculates the x-coordinate of the vertex. Plugging this x-value back into the original quadratic function helps find the y-coordinate.

For example, with the function \( f(x) = x^2 - x + 5 \), identify coefficients as , , and . Then use the vertex formula: \( x = -\frac{-1}{2\cdot1} = \frac{1}{2} \). Once you have the x-coordinate, substitute it back into the function to find the y-coordinate: \[ f\left( \frac{1}{2} \right) = \left( \frac{1}{2} \right)^2 - \frac{1}{2} + 5 = \frac{19}{4} \]. The vertex is at \( \left( \frac{1}{2}, \frac{19}{4} \right) \).

coefficients

Coefficients are numbers that multiply the variables in polynomial functions. In a quadratic function, the coefficients are a, b, and c in \( f(x) = ax^2 + bx + c \).

• 
  is the leading coefficient and determines the parabola's direction and width. If is positive, the parabola opens upwards. If it's negative, the parabola opens downwards. A larger absolute value of means the parabola is narrower, while a smaller absolute value means it's wider.


• 
  is the coefficient of the linear term and influences the parabola's axis of symmetry location. Alongside , helps determine the x-coordinate for the vertex using the formula: \( x = -\frac{b}{2a} \).


• 
  is the constant term and determines where the parabola crosses the y-axis.

Understanding the roles of these coefficients helps in graphing the quadratic function and finding its key features, like the vertex, axis of symmetry, and direction.