Problem 44 - College Algebra with Corequisite Support

Question

For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the \(x\) -intercepts) by using \(2^{\text {nd }}\) CALC 2:zero. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth. \(\mathrm{Y}_{1}=4 x^{2}+3 x-2\)

Step-by-Step Solution

Answer

The x-intercepts are approximately \(x = -1.000\) and \(x = 0.500\).

Step 1: Enter the Expression

Turn on your graphing calculator and navigate to the 'Y=' menu. Enter the given expression \( Y_1 = 4x^2 + 3x - 2 \) into the calculator as the function to graph.

Step 2: View the Graph

Press 'GRAPH' to view the graph of the equation. You should see a parabola intersecting the x-axis, indicating potential zeroes of the function.

Step 3: Access the Zero-CALC Function

Press the '2nd' button followed by the 'TRACE' button to access the 'CALC' menu. Select option '2:zero' by either pressing '2' or scrolling and hitting 'ENTER'. This tool allows you to find the x-intercepts of the graph.

Step 4: Set the Left Bound

Using the arrow keys, move the cursor close to one of the zeroes of the parabola, ensuring the cursor is to the left of where the parabola crosses the x-axis. Press 'ENTER' to set the left bound.

Step 5: Set the Right Bound

Now, move the cursor using the arrow keys so that it's positioned to the right of the zero you are trying to find. Press 'ENTER' to set the right bound.

Step 6: Make a Guess

Move the cursor between the bounds, preferably close to where the zero seems to be. Press 'ENTER' for the calculator to make a guess and calculate the zero.

Step 7: Record the First Zero

The calculator will display the x-coordinate of the first zero on the screen. Note this value down. Round it to the nearest thousandth.

Step 8: Repeat for the Other Zero

Repeat Steps 3-7 to find the second zero of the function by setting new left and right bounds on a different x-intercept.

Step 9: Record the Second Zero

Note down the x-coordinate of the second zero. Ensure this value is also rounded to the nearest thousandth.

Key Concepts

Graphing Utility Techniques

Graphing utilities, like calculators, can be powerful tools for visualizing equations and functions. To find the zeroes of a quadratic equation, we use these devices to create a visual representation of the equation in the form of a graph. Here's how you can effectively use your graphing calculator:

• Entering an Expression: Start by entering the equation in the calculator's function menu. For the given exercise, input the quadratic function, for example, \(Y_1 = 4x^2 + 3x - 2\).
• Viewing the Graph: Upon pressing 'GRAPH,' you can see the plot of your function. Focus on the x-axis where the curves of the parabola intersect, indicating potential zeroes.
• Utilizing CALC Tools: Use the '2:zero' feature in the 'CALC' menu to find specific zeroes. This feature works best by setting left and right bounds around the suspected zero and letting the calculator make a precise guess.

These steps simplify the process of isolating zeroes in quadratic functions and validate the analytical solutions.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). They represent parabolas when graphed, with their degree resulting in at most two x-intercepts or zeroes. Understanding the properties of these equations is crucial:

• Standard Form: Typically, quadratic equations are expressed in the standard form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
• Parabolic Shape: The graph is a parabola that opens upwards if \(a > 0\) and downwards if \(a < 0\).
• Finding Zeroes: Algebraically, zeroes are found using factoring, completing the square, or the quadratic formula. Graphically, a graphing calculator can aid in pinpointing these zeroes effortlessly.

By mastering both the algebraic and graphical methods, students can fully comprehend how quadratic functions behave.

Parabola Intersections

A parabola's intersections with the x-axis are pivotal because they represent the zeroes of the function or the solutions to the quadratic equation. Here's what you need to know:

• X-Intercepts: When a parabola crosses the x-axis, the related \(y\) value is zero. Thus, the x-coordinate at this intersection is a solution to the equation.
• Intersection Method: To discover intersections, graph the parabola using a calculator and employ the 'zero' function to find where the curve meets the axis.
• Practical Relevance: In many real-world applications, solutions to quadratic equations offer vital information such as maximum heights or optimal points in optimization problems.

Through practice, understanding how parabolas intersect with the x-axis becomes an intuitive task.

Rounding Solutions

Once you have the solutions to a quadratic equation from your graphing utility, it is important to refine them by rounding. Rounding to the nearest thousandth ensures precision while facilitating easier interpretation and communication of results.

• Calculator Output: Graphing calculators provide precise solutions, sometimes in long decimal formats.
• Rounding Process: Identify the thousandth place in the number. Look at the digit immediately after it; if it is 5 or greater, round up. Otherwise, round down.
• Application in Exercises: In the provided exercise, make careful use of rounding after identifying the zeroes, ensuring communication of results is clear and consistent.
• Accuracy and Precision: Rounding balances the need for clarity with the practical requirement of precision.

Getting comfortable with rounding techniques is vital to delivering accurate and user-friendly solutions.