Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, rewrite the quadratic functions in standard form and give the vertex. $$ f(x)=3 x^{2}-5 x-1 $$

Step-by-Step Solution

Answer

The function is already in standard form, and the vertex is at \( \left( \frac{5}{6}, -\frac{37}{12} \right) \).

Step 1: Identify the Quadratic Coefficients

For the quadratic function given by \( f(x) = 3x^2 - 5x - 1 \), identify the coefficients \( a \), \( b \), and \( c \) from the standard quadratic form \( ax^2 + bx + c \). Here, \( a = 3 \), \( b = -5 \), and \( c = -1 \).

Step 2: Find the Vertex Formula for x-coordinate

To find the vertex of the quadratic function, use the vertex formula for the x-coordinate: \( x = -\frac{b}{2a} \). Substitute \( b = -5 \) and \( a = 3 \) to find: \[ x = -\frac{-5}{2 \times 3} = \frac{5}{6} \].

Step 3: Calculate the y-coordinate of the Vertex

Substitute the x-coordinate \( x = \frac{5}{6} \) back into the function \( f(x) = 3x^2 - 5x - 1 \) to find the y-coordinate of the vertex. Compute:\[ f\left(\frac{5}{6}\right) = 3\left(\frac{5}{6}\right)^2 - 5\left(\frac{5}{6}\right) - 1 \]Continue calculation:\[ = 3 \times \frac{25}{36} - \frac{25}{6} - 1 \]\[ = \frac{75}{36} - \frac{150}{36} - \frac{36}{36} \]\[ = \frac{75 - 150 - 36}{36} \]\[ = \frac{-111}{36} \]\[ = -\frac{37}{12} \].

Step 4: Identify the Vertex

The vertex coordinates are \( \left( \frac{5}{6}, -\frac{37}{12} \right) \), derived from our calculations in steps 2 and 3.

Key Concepts

Vertex

The vertex of a quadratic function is a crucial point that gives us valuable insight into the graph of the parabola. It represents the highest or lowest point of the graph, depending on the direction of the parabola's opening. For the function \[ f(x) = 3x^2 - 5x - 1 \]to find the vertex, we use the formula for the x-coordinate:\[ x = -\frac{b}{2a} \]Inserting the values from our quadratic coefficients, \( b = -5 \) and \( a = 3 \), we find the x-coordinate of the vertex as\[ x = \frac{5}{6} \]After finding the x-coordinate, plug this value back into the original function to find the y-coordinate:\[ f\left(\frac{5}{6}\right) = 3\left(\frac{5}{6}\right)^2 - 5\left(\frac{5}{6}\right) - 1 \]This calculation shows that the y-coordinate is\[ -\frac{37}{12} \]Leaving us with the vertex\[ \left( \frac{5}{6}, -\frac{37}{12} \right) \].You can always rely on the vertex to help you understand the direction of a quadratic graph. Remember:

• If the parabola opens upwards (if \( a > 0 \)), the vertex is the lowest point.
• If the parabola opens downwards (if \( a < 0 \)), the vertex is the highest point.

Standard Form

Quadratic functions are often written in standard form. This standard format is easy to recognize and manipulate. The standard form appears as:\[ ax^2 + bx + c \]where \( a \), \( b \), and \( c \) are constants. The 'standard form' is extremely handy for quickly identifying a few important aspects:

• Direction of the parabola's opening (upwards or downwards) is determined by the sign of coefficient \( a \).
• The quadratic function in standard form is always a parabola when plotted on a graph.

To transform any quadratic into its standard form, simply rearrange and identify \( a \), \( b \), and \( c \) from your equation.
For example: In \( f(x) = 3x^2 - 5x - 1 \), \( a = 3 \), \( b = -5 \), and \( c = -1 \).Standard form is often the starting block for further manipulations, such as finding intercepts or converting to vertex form.Remember:

• Standard form is the key to many basic and advanced calculus applications.
• It allows us to promptly proceed with calculations to find the vertex, intercepts, and even roots using various algebraic techniques.

Quadratic Coefficients

Quadratic coefficients are the constants \( a \), \( b \), and \( c \) in the quadratic expression \[ ax^2 + bx + c \].Each of these has its own role in shaping the graph of a quadratic equation.

Understanding the Coefficients:

• \( a \): Determines the direction of the parabola (upward if positive, downward if negative) and its "width" (the larger \( a \) is, the "narrower" the parabola).
• \( b \): Affects the symmetry and the location of the vertex. It helps in calculating the x-coordinate of the vertex using the formula \(-\frac{b}{2a}\).
• \( c \): The constant term which denotes the y-intercept, showing where the parabola intersects the y-axis when \( x = 0 \).

In our example of \( f(x) = 3x^2 - 5x - 1 \), the coefficients are \( a = 3 \), \( b = -5 \), and \( c = -1 \). Each plays a significant role in shaping the graph. Understanding these coefficients lets you predict and sketch graphs even before precise calculations.