Problem 30 - 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. \(f(x)=2 x^{2}-x+1\)

Step-by-Step Solution

Answer

The vertex of the function \(f(x)=2x^{2}-x+1\) is at the point (1/4, 7/8). The graph of the function is an upward opening parabola with the vertex at this point.

Step 1: Calculate Vertex X-Coordinate

The x-coordinate of the vertex can be found by the formula \(-b/2a\). In this function, a=2 and b=-1. Plugging these into the formula gives \(-(-1)/2(2)=1/4.\

Step 2: Calculate Vertex Y-Coordinate

The y-coordinate of the vertex is found by plugging the x-value into the function. Doing this gives \(f(1/4)=2(1/4)^{2}-1/4+1 = 2/16 - 1/4 + 1 = 1/8 - 1/4 + 1 = 7/8.\

Step 3: Combine Vertex Coordinates

Therefore, the vertex of the function is at the point (1/4, 7/8).

Step 4: Graphing the Function

Use a graphing utility like a graphing calculator or a software program to graph the equation \(f(x) = 2x^{2} - x + 1\). The parabola will have a vertex at the point (1/4, 7/8) and will open upward, corresponding with the positive co-efficient in front of \(x^{2}\). This should verify the previous manual results.

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point, depending on the direction the parabola opens. For the quadratic function in standard form, \(f(x) = ax^2 + bx + c\), the vertex can be found using a simple formula. The x-coordinate of the vertex is given by \(-\frac{b}{2a}\). This formula helps locate the line of symmetry of the parabola, dividing it into two mirror-image halves.

To calculate the y-coordinate, substitute the x-coordinate back into the function. This gives you the specific point on the graph, fully determining the vertex. In the example with \(f(x)=2x^2-x+1\), the vertex is found to be at \(\left(\frac{1}{4}, \frac{7}{8}\right)\). This point represents the minimum value of the function, as the parabola opens upwards.

Finding the vertex is essential for understanding the shape and direction of a parabola, aiding in graphing and analysis.

Graphing Parabolas

Graphing a parabola involves understanding its direction, width, and position. Once the vertex is identified, you have a starting point. The next step is to determine the direction in which the parabola opens. This is influenced by the "a" coefficient in the quadratic equation. If \(a > 0\), the parabola opens upwards. If \(a < 0\), it opens downwards.

With \(f(x) = 2x^2 - x + 1\), the graph opens upward because \(a = 2\) is positive. This tells us that the vertex represents the lowest point on the graph, known as the minimum.

Using a graphing utility can greatly assist by visually confirming the parabola's features. Check symmetry around the vertex, and observe the parabola's width. A larger \(a\) value means a narrower graph, while a smaller \(a\) causes a wider graph. Always verify the manual calculations with a graphing tool to ensure accuracy in your sketch.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is expressed as \(ax^2 + bx + c\). Each parameter \(a\), \(b\), and \(c\) has a specific role in shaping the graph of the function, known as a parabola.

- "a" affects the direction and width of the parabola. Positive values of \(a\) mean the parabola opens upwards, while negative values open it downwards. The magnitude of \(a\) affects the "stretch" or "narrowness."- "b" influences the position of the vertex along the x-axis, playing a role in symmetry.- "c" represents the y-intercept, showing where the graph crosses the y-axis.

Understanding these components allows you to predict the graph's appearance straight from the equation. In the function \(f(x) = 2x^2 - x + 1\), knowing each part assists in predicting the vertex position, opening direction, and overall shape. This knowledge builds a foundation to efficiently tackle more complex quadratic functions.