Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=x^{2}+3 x+\frac{1}{4}\)

Step-by-Step Solution

Answer

The vertex of the function \(f(x)=x^{2}+3x+\frac{1}{4}\) is \((-1.5, -2.25)\)

Step 1: Identify the coefficients

In the given quadratic function \(f(x)=x^{2}+3x+\frac{1}{4}\), the coefficients are \(a=1\), \(b=3\) and \(c=\frac{1}{4}\).

Step 2: Calculate the x-coordinate of the vertex

The x-coordinate of the vertex of a parabola is given by \(-\frac{b}{2a}\). Substituting the given values of \(a\) and \(b\), we get \(-\frac{3}{2*1}=-1.5\). So, the x-coordinate of the vertex is -1.5.

Step 3: Calculate the y-coordinate of the vertex

Substitute the x-coordinate of the vertex into the function \(f(x)\) to find the y-coordinate. This makes \(f(-1.5)=(-1.5)^{2}+3(-1.5)+\frac{1}{4}= -2.25\). So, the y-coordinate of the vertex is -2.25.

Step 4: Identify the vertex

The vertex of the parabola is the point \((-1.5, -2.25)\)

Key Concepts

Graphing Quadratic Functions

Graphing a quadratic function is a key skill in algebra that provides a visual representation of the equation's behavior. A quadratic function is typically written in the form f(x) = ax² + bx + c. The graph of every quadratic function is a parabola, a U-shaped curve that can open upwards or downwards.

When graphing f(x) = x² + 3x + 1/4, one would usually start by finding the y-intercept by setting x to 0. This gives the point (0, c), which is (0, 1/4) for our function. After plotting the y-intercept, the next step is to find the vertex, which is the highest or lowest point on the graph, depending on whether the parabola opens up or down. Since a = 1 is positive, our parabola opens upward, and the vertex will be a minimum point.

Using symmetrical properties of the parabola, additional points can be plotted by choosing values for x and calculating the corresponding y values. Then, sketch the parabola by connecting these points smoothly. To ensure the graph's accuracy, one could use a graphing utility or software. This helps to verify that the plotted points and drawn parabola correctly represent the quadratic function's equation.

Finding the Vertex of a Parabola

The vertex of a parabola represents the peak or apex of the curve. For a quadratic function f(x) = ax² + bx + c, the coordinates of the vertex can be determined algebraically. The x-coordinate is given by the formula x = -b/(2a).

For our function f(x) = x² + 3x + 1/4, we already have the coefficients a = 1 and b = 3. Applying them to the formula, we get the x-coordinate of the vertex as -1.5. To find the y-coordinate, we substitute this x-coordinate back into the function: f(-1.5) = (-1.5)² + 3(-1.5) + 1/4, which simplifies to -2.25. Therefore, the vertex of this parabola is at the point (-1.5, -2.25).

Knowing the vertex is crucial because it helps in graphing the parabola and is also used in various applications, such as determining the maximum height of a projectile or the minimum cost in a business model.

Coefficients in Quadratic Equations

Coefficients in a quadratic equation f(x) = ax² + bx + c play significant roles in determining the graph's shape and position. The coefficient a influences the direction in which the parabola opens as well as its width. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. A larger absolute value of a makes the parabola steeper, while a smaller absolute value makes it wider.

The coefficient b affects the position of the vertex and, consequently, the entire parabola along the horizontal axis. Lastly, the coefficient c represents the y-intercept, the point where the graph crosses the y-axis. In our example, with a = 1, b = 3, and c = 1/4, we can predict that the parabola opens upwards and is relatively wide, and the y-intercept is at 0.25.

Understanding these coefficients is key not only in graphing but also in comprehending the quadratic function's underlying behavior, enabling us to solve real-world problems effectively.