Problem 45 - College Algebra

Question

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. $$ \mathrm{Y}_{1}=-3 x^{2}+8 x-1 $$

Step-by-Step Solution

Answer

The zeroes of the equation rounded to the nearest thousandth are \( x \approx 0.135 \) and \( x \approx 2.468 \).

Step 1: Graph the Equation

Enter the function \( Y_1 = -3x^2 + 8x - 1 \) into your graphing utility. Make sure to correctly input the expression by starting with the square term \(-3x^2\), followed by \(+8x\), and ending with \(-1\).

Step 2: Access the Zero Finder

Once the graph is displayed, access the zero finding feature by pressing the \(2^{nd} \) button followed by the CALC button on your graphing utility. Then, select the option number 2, which is 'zero.'

Step 3: Set the Left Bound

Move the cursor to just left of where the graph crosses the x-axis. Press Enter to set this position as the left bound.

Step 4: Set the Right Bound

Move the cursor to just right of where the graph appears to cross the x-axis. Press Enter to set this point as the right bound.

Step 5: Guess the Zero

Move the cursor close to the x-intercept between the bounds you've set. Press Enter to have the calculator guess the zero. Note down the x-intercept, which is our zero of the function.

Step 6: Round Result to Nearest Thousandth

Look at the x-intercept value found by the calculator and round it to the nearest thousandth if necessary. Repeat the process for each x-intercept if there are multiple.

Key Concepts

Finding Zeros

When you are tasked with finding the zeros of a function using a graphing calculator, you're essentially looking for the points where the graph touches or crosses the x-axis. These points are known as zeros or roots of the function. For quadratic equations, there can be up to two zeros.

• Use the calculator's "zero" function.
• Set left and right bounds around the x-intercept area.
• Let the calculator guess the precise location of the zero.

By finding these x-intercepts, you're identifying the solutions to the equation when it is set to zero.

Quadratic Equations

Quadratic equations are polynomial equations of degree two. Their standard form is: \[ax^2 + bx + c = 0\]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The equation represents a parabola when graphed. It can open upwards or downwards depending on the sign of \(a\).

• If \(a > 0\), the parabola opens upwards.
• If \(a < 0\), it opens downwards.

The shape and direction of the parabola will influence how you find the x-intercepts or zeros of the equation.

X-intercepts

X-intercepts, or zeros, are the points where the graph of the equation crosses the x-axis. For quadratic equations, the x-intercepts also signify the solution to the equation when \(Y=0\). Using a graphing calculator makes finding these points much simpler.

• Input the equation into the graphing calculator.
• Use the graphing calculator to zoom and adjust the view for a more precise location of the intercepts.
• Find the intercepts using the 'zero' function.

Once the calculator estimates where the curve intersects the x-axis, those estimated points are your x-intercepts.

Rounding Numbers

Rounding numbers is an essential mathematical skill, often required when zeros do not appear as simple integers. When rounding numbers to the nearest thousandth:

• Look at the fourth digit after the decimal point.
• If it's 5 or greater, round up the third digit.
• If it's less than 5, keep the third digit as it is.

For example, if your calculator gives an x-intercept of 2.4567, you would round this to 2.457. This precision is crucial for accurate and consistent results.