Problem 78
Question
Use a graphing utility to graph the function and determine any \(x\) -intercepts. Set \(y=0\) and solve the resulting equation to confirm your result. $$y=2 x-1+\frac{1}{x-2}$$
Step-by-Step Solution
Verified Answer
The x-intercepts for the function \(y=2x-1+\frac{1}{x-2}\) are \(x=\frac{2}{3}\) and \(x=3\).
1Step 1: Set the equation to 0
Setting \(y = 0\) gives: \[0=2x-1+\frac{1}{x-2}\]
2Step 2: Solve the equation
Solve the resulting equation for \(x\). This will involve moving the terms to one side to compute for \(x\) and get the x-intercepts. Multiply through by \(x-2\) to remove the fraction: \[0=2x(x-2)-1(x-2)+(x-2)(x-2)\] Then simplify : \[0 = 2x^2 -4x -x +2 +x^2 -4x +4\] Further simplifying gives: \[0 = 3x^2-8x+6\] Solving this by factoring, we get : \[(3x-2)(x-3) = 0\]. Thus, \(x = \frac{2}{3}\) and \(x = 3\).
3Step 3: Use a graphing utility
Enter the function \(y=2x-1+\frac{1}{x-2}\) into the graphing utility. Check that the x values obtained in the previous step are where the function crosses the x-axis. This serves as a confirmation.
Key Concepts
Solving Rational EquationsFactoring Quadratic EquationsUsing Technology in AlgebraX-axis Intercepts
Solving Rational Equations
A rational equation is one that contains a ratio of two polynomials, similar to the function \(y=2x-1+\frac{1}{x-2}\) from our exercise. To solve such equations, the goal is to find the value of the variable that makes the equation true.
When faced with a rational equation, setting the equation equal to zero allows us to identify the points where the function intersects the x-axis, as these are the points for which \(y=0\). To solve the equation, we often aim to clear the denominator to avoid division by zero, a fundamental restriction in algebra. In our example, we multiply through by \(x-2\) to remove the complex fraction. After simplifying, we find ourselves with a quadratic equation that can be factored further to find the x-intercepts.
When faced with a rational equation, setting the equation equal to zero allows us to identify the points where the function intersects the x-axis, as these are the points for which \(y=0\). To solve the equation, we often aim to clear the denominator to avoid division by zero, a fundamental restriction in algebra. In our example, we multiply through by \(x-2\) to remove the complex fraction. After simplifying, we find ourselves with a quadratic equation that can be factored further to find the x-intercepts.
Factoring Quadratic Equations
Factoring is a crucial skill when it comes to solving quadratic equations, and it involves breaking down a complex expression into simpler ones that multiply together to give the original expression.
In our example, we simplified the rational equation to \(3x^2-8x+6=0\), which is a quadratic equation. We then factored this equation to \(3x-2\) times \(x-3\) equals zero. After factoring, we apply the Zero Product Property, which states that if a product of two expressions is zero, then at least one of the expressions must be zero. This leads us to the solutions \(x=\frac{2}{3}\) and \(x=3\), corresponding to where the graph of our function crosses the x-axis.
In our example, we simplified the rational equation to \(3x^2-8x+6=0\), which is a quadratic equation. We then factored this equation to \(3x-2\) times \(x-3\) equals zero. After factoring, we apply the Zero Product Property, which states that if a product of two expressions is zero, then at least one of the expressions must be zero. This leads us to the solutions \(x=\frac{2}{3}\) and \(x=3\), corresponding to where the graph of our function crosses the x-axis.
Using Technology in Algebra
Technology can be a powerful tool in algebra, helping visualize functions and verify calculations. Graphing utilities, like a graphing calculator or computer software, allow us to plot equations quickly and precisely.
When graphing the function \(y=2x-1+\frac{1}{x-2}\), we can use a graphing utility to visually confirm our calculated x-intercepts. By entering the function into the utility and analyzing the graph, we can check if the function indeed intersects the x-axis at the points we found algebraically, offering a reliable way to validate our work.
When graphing the function \(y=2x-1+\frac{1}{x-2}\), we can use a graphing utility to visually confirm our calculated x-intercepts. By entering the function into the utility and analyzing the graph, we can check if the function indeed intersects the x-axis at the points we found algebraically, offering a reliable way to validate our work.
X-axis Intercepts
The x-axis intercepts, also known as zeros or roots of the function, are the points where the graph of the function crosses the x-axis. These are found by solving the equation \(y=0\), which gives us the values of \(x\) when the output of the function is zero.
In the context of our example, once we factored the quadratic equation to \(3x-2\) and \(x-3\), we identified the x-axis intercepts as \(x=\frac{2}{3}\) and \(x=3\). It's crucial to understand that these intercepts represent the values of \(x\) for which the entire function has no height, which in graphical terms, means the curve touches the x-axis.
In the context of our example, once we factored the quadratic equation to \(3x-2\) and \(x-3\), we identified the x-axis intercepts as \(x=\frac{2}{3}\) and \(x=3\). It's crucial to understand that these intercepts represent the values of \(x\) for which the entire function has no height, which in graphical terms, means the curve touches the x-axis.
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