Problem 11 - Technical Mathematics with Calculus

Question

Graph each function. Set the viewing window for \(x\) and \(y\) initially from -5 to 5 then resize if needed. $$y=1.74 x^{2}-2.35 x+1.84$$

Step-by-Step Solution

Answer

The vertex of the parabola can be found at approximately x = 0.676, and the corresponding y-value can be calculated by substituting x back into the original equation. Plot the vertex, axis of symmetry, additional points, and then sketch the parabola on the graph.

Step 1: Plot the Vertex

First, find the vertex of the parabola. The equation is in the form of a quadratic function: \( y = ax^2 + bx + c \). To find the vertex, use the formula \( x = -\frac{b}{2a} \). For the given equation \( y = 1.74x^2 - 2.35x + 1.84 \), the vertex will be at \( x = -\frac{-2.35}{2 \times 1.74} \).

Step 2: Calculate the Vertex

Compute the x-coordinate of the vertex using the formula from Step 1: \( x = -\frac{-2.35}{2 \times 1.74} \). Simplify to find the x-coordinate of the vertex, and then substitute this x-coordinate back into the original equation to find the corresponding y-coordinate.

Step 3: Plot the Axis of Symmetry

The line of symmetry will be a vertical line that passes through the x-coordinate of the vertex. Plot this line on the graph to help in sketching the parabola.

Step 4: Plot Additional Points

Choose x-values around the vertex and compute the corresponding y-values, then plot these points on the graph. This will give the shape of the parabola around the vertex.

Step 5: Draw the Parabola

Using the vertex, the axis of symmetry, and the points computed in Step 4, draw a smooth curve through these points to complete the parabola. Adjust the viewing window if the full shape of the parabola isn't clear.

Key Concepts

Plotting Parabolas

Graphing parabolas is a fundamental skill in algebra that allows students to visualize quadratic functions. When plotting a parabola, it's important to start by identifying essential points and lines that define its shape. These include the vertex, the axis of symmetry, and several points on either side of the vertex.

To plot a parabola accurately, one should begin by calculating the vertex, which is the highest or lowest point on the graph, depending on the direction the parabola opens. Then, identify the axis of symmetry, a vertical line that runs through the vertex and divides the parabola into mirror images. After these two steps, additional points are computed by substituting values for x into the equation, and their corresponding y values are plotted. Lastly, connect these points with a smooth, continuous curve to form the parabola, making sure the curve passes through the vertex and is symmetrical about the axis of symmetry. Adjusting the viewing window is crucial here, as it ensures that the entire parabola is visible and accurately represented on the graph.

Vertex of a Parabola

The vertex of a parabola is a key feature that helps to define its precise shape and location on a graph. It is the point where the parabola changes direction, marking the peak or the trough of the curve. For a parabola described by the equation in the vertex form, the vertex can be extracted directly. However, for the standard quadratic equation form, such as the one in our exercise (\(y = ax^2 + bx + c\)), we use the formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex.

Once we have the x-coordinate, it's substituted back into the original equation to find the corresponding y-coordinate, completing the coordinates of the vertex. This point is essential to plot first as it serves as a reference for sketching the rest of the parabola. Remember that the vertex is a maximum if the parabola opens downwards (a < 0) and a minimum if it opens upwards (a > 0).

Axis of Symmetry

The axis of symmetry in the context of a parabola is a vertical line that divides the graph of the quadratic function into two mirror-image halves. It is a key concept that ensures the parabola is symmetric. The equation of the axis of symmetry for a parabola given by \( y = ax^2 + bx + c \) can be found using the same formula used to locate the vertex's x-coordinate, \( x = -\frac{b}{2a} \).

This line of symmetry passes through the vertex, and every point on one side of the axis has a corresponding point with the same distance from the axis on the other side. When graphing, it is beneficial to plot the axis of symmetry before plotting the additional points, as it acts as a guide for ensuring that points are reflected correctly across the graph. The symmetry about this line means that if you have a point on one side of the parabola, there's a matching point on the other side that helps give the parabola its characteristic shape.