Problem 7 - Algebra A Combined Function

Question

Find the vertex of the graph of each quadratic function by completing the square or using the vertex formula. $$ f(x)=-x^{2}+x+1 $$

Step-by-Step Solution

Answer

The vertex is \(\left( \frac{1}{2}, \frac{5}{4} \right)\).

Step 1: Identify the Quadratic Function

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

Step 2: Apply the Vertex Formula

The vertex of a quadratic function \( f(x) = ax^2 + bx + c \) can be found using the vertex formula \( x = -\frac{b}{2a} \). For our function, substitute \( a = -1 \) and \( b = 1 \) into the formula: \[ x = -\frac{1}{2(-1)} = \frac{1}{2} \].

Step 3: Calculate the y-coordinate of the Vertex

Substitute \( x = \frac{1}{2} \) back into the original function to find \( y \): \[ f\left(\frac{1}{2}\right) = -\left(\frac{1}{2}\right)^2 + \frac{1}{2} + 1 \]. This simplifies to \[ f\left(\frac{1}{2}\right) = -\frac{1}{4} + \frac{1}{2} + 1 = -\frac{1}{4} + \frac{2}{4} + \frac{4}{4} = \frac{5}{4} \].

Step 4: Present the Vertex

The vertex of the quadratic function is at \( \left( \frac{1}{2}, \frac{5}{4} \right) \).

Key Concepts

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial, thus making it easier to identify the vertex of a quadratic function. Consider a quadratic function in the form of \[ f(x) = ax^2 + bx + c \].Here's a brief overview of completing the square:

• First, ensure the coefficient of \(x^2\) is 1. If it isn't, factor it out from the \(x^2\) and \(x\) terms.
• Next, take the coefficient of \(x\), divide it by 2, and square it. This number is added and subtracted within the expression to complete the square.
• Rewrite the quadratic expression as a square of a binomial and adjust the constant term as necessary.

Completing the square is useful not only for finding the vertex but also for solving quadratic equations and analyzing the properties of quadratic graphs like identifying maximum or minimum points.

Vertex Formula

The vertex formula provides a quick method to find the x-coordinate of the vertex of a quadratic function. When a quadratic function is in the standard form \( ax^2 + bx + c \), the x-coordinate of the vertex can be calculated using the formula:\[ x = -\frac{b}{2a} \].This formula is derived from completing the square and is extremely useful as it provides direct access to one critical feature of the quadratic's graph. Once the x-coordinate is found, substitute this value back into the original function to get the y-coordinate. This gives us the complete vertex point.Using the vertex formula simplifies finding the vertex when compared to methods such as graphing or completing the square, especially when dealing with more complex coefficients.

Quadratic Function in Standard Form

A quadratic function in its standard form is expressed as\[ f(x) = ax^2 + bx + c \]. Here, \(a\), \(b\), and \(c\) are constants, with \(aeq0\). The standard form is quite versatile as it allows for easy application of the vertex formula and other algebraic manipulations such as completing the square.Some key points regarding the standard form include:

• The coefficient \(a\) determines the direction of the parabola's opening. If \(a\) is positive, the parabola opens upwards, making the vertex a minimum. Conversely, if \(a\) is negative, the parabola opens downward, presenting the vertex as a maximum.
• The constants \(b\) and \(c\) affect the parabola's axis of symmetry and vertical shift, respectively.

Understanding the standard form of a quadratic d function lays the groundwork for further exploration into its roots, axis of symmetry, and extremum points.