Problem 48 - College Algebra with Corequisite Support

Question

To solve the quadratic equation \(0.3 x^{2}+2 x-4=2,\) we can graph these two equations $$\begin{array}{l} Y_{1}=0.3 x^{2}+2 x-4 \\ Y_{2}=2 \end{array}$$ and find the points of intersection. Recall \(2^{\text {nd }}\) CALC 5:intersection. Do this and find the solutions to the nearest tenth.

Step-by-Step Solution

Answer

The solutions are approximately \(x = -0.7\) and \(x = 4.7\).

Step 1: Rearrange Equations

First, we rewrite the quadratic equation from the problem statement for graphing: \(Y_1 = 0.3x^2 + 2x - 4\) and \(Y_2 = 2\). These will be the equations we graph to find their points of intersection.

Step 2: Graph the Equations

Plot the graph of \(Y_1 = 0.3x^2 + 2x - 4\) and \(Y_2 = 2\) on the coordinate plane. This can typically be done using a graphing calculator or software.

Step 3: Identify Intersection Points

Use the 'intersect' function, often found under the 'calc' menu of graphing calculators, to find the x-values where \(Y_1\) and \(Y_2\) intersect. This involves selecting the two curves and letting the calculator find the points of intersection.

Step 4: Record the Solutions

The x-values obtained from the intersection points are the solutions to the equation \(0.3x^2 + 2x - 6 = 0\). These values represent where the quadratic curve intersects the horizontal line \(Y = 2\).

Step 5: Round to the Nearest Tenth

Round the x-values obtained from the intersection to the nearest tenth to provide the final answer.

Key Concepts

Graphing Quadratic Functions

Quadratic functions are polynomials of the form \(ax^2 + bx + c\). These equations create a U-shaped graph known as a parabola. The graph can face upwards or downwards depending on the sign of the coefficient \(a\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

To graph a quadratic function like \(Y_1 = 0.3x^2 + 2x - 4\), follow these steps:

• Find the vertex, which is the highest or lowest point of the graph. Use the formula \(x = -\frac{b}{2a}\) to find the x-value of the vertex.
• Determine the y-intercept by setting \(x = 0\) in the equation.
• Identify additional points by substituting other values for \(x\) and calculating the corresponding \(Y_1\).
• Draw the symmetric nature of the parabola using these points.

Graphing provides a visual representation of the equation, helping identify solutions through intersection with other graphs, such as a horizontal line.

Intersection of Graphs

Finding the intersection of graphs, such as a parabolic curve and a horizontal line, involves determining where the two graphs meet. This process is crucial for solving equations graphically, such as finding the solution to \(0.3x^2 + 2x - 4 = 2\).

Here's how to locate the intersections:

• Graph both equations, \(Y_1 = 0.3x^2 + 2x - 4\) and \(Y_2 = 2\), on the same coordinate plane.
• Look for points where the parabolic curve (\(Y_1\)) crosses the horizontal line (\(Y_2\)).
• These intersection points correspond to x-values that satisfy the equation.

To find precise intersection points, a graphing calculator can be used to apply calculus tools or numerical methods provided by the calculator interface.

Using Graphing Calculators

Graphing calculators are essential tools for visualizing and solving quadratic equations graphically. They make plotting functions and finding intersections easier and more accurate.

To use a graphing calculator for finding intersection points, follow these steps:

• First, input the equations to be graphed, such as \(Y_1 = 0.3x^2 + 2x - 4\) and \(Y_2 = 2\).
• Select the graph function and ensure both equations are plotted on the screen.
• Utilize the "intersection" tool, typically found in the "calc" menu. This tool helps you select the two graphs and calculates the exact x-values where they intersect.
• Use the calculator's cursor or arrow keys to pinpoint the intersection, then confirm with the "enter" key to obtain the values.

Graphing calculators not only facilitate the graphical approach but also ensure that solutions are accurate to the decimal point, which is helpful when rounding to the nearest tenth or any specified precision.