Problem 38
Question
(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(g(x)=5 x^{2}-10 x-5\)
Step-by-Step Solution
Verified Answer
By solving the equation algebraically, you get the zeros of the function. Then by graphing the function, you visually see the zeros as the points where the graph intersects the x-axis. Upon comparing, you will find that the results match.
1Step 1: Setting Equation to Zero
Find the zeros of the function algebraically by setting the function \(g(x)=5x^{2}-10x-5\)equal to zero. The equation becomes \(5x^{2}-10x-5=0\)
2Step 2: Solving the Quadratic Equation
Use the quadratic formula to solve for x. The quadratic formula is \(x=[-b±sqrt(b^{2}-4ac)]/2a\), where a, b, and c are coefficients of the quadratic equation and 'sqrt' denotes square root. In this equation, a=5, b=-10, and c=-5. Plugging these values into the formula gives x values, these x values are the zeros of the function.
3Step 3: Graphing the Function
Use a graphing tool to plot the function \(g(x)=5x^{2}-10x-5\). The point where the graph intersects the x-axis are the zeros of the function.
4Step 4: Comparing the Results
Compare the x-values obtained in Step 2 to the x-coordinates of the intercepts on the graph from Step 3. If the results from the graphical and algebraic solutions match, then the solutions have been found correctly.
Key Concepts
Zero of a FunctionQuadratic FormulaGraphing Quadratic Functions
Zero of a Function
The concept of the 'zero of a function' is a fundamental building block in understanding various mathematical and real-world phenomena. In simple terms, a zero of a function is a point where the function crosses or touches the x-axis on a graph. At this point, the function's value is zero hence the term 'zero of a function'.
When we discuss quadratic functions, finding the zeros can often solve a variety of problems, such as determining the roots of an equation. In our exercise, we are interested in finding the zeros of the function \(g(x)=5x^{2}-10x-5\) algebraically. By setting the function equal to zero, we form the equation \(5x^{2}-10x-5=0\), which we can then solve to find the values of \(x\) that make the function zero. These values are crucial as they often represent key points in the context of the problem, such as the timing of an event in a physics problem or break-even points in economics.
When we discuss quadratic functions, finding the zeros can often solve a variety of problems, such as determining the roots of an equation. In our exercise, we are interested in finding the zeros of the function \(g(x)=5x^{2}-10x-5\) algebraically. By setting the function equal to zero, we form the equation \(5x^{2}-10x-5=0\), which we can then solve to find the values of \(x\) that make the function zero. These values are crucial as they often represent key points in the context of the problem, such as the timing of an event in a physics problem or break-even points in economics.
Quadratic Formula
The quadratic formula is an essential tool for solving quadratic equations, represented by \(ax^{2} + bx + c = 0\). It provides a systematic approach to finding the zeros of quadratic functions.
The quadratic formula is:\[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]
It utilises the coefficients of the quadratic equation: \(a\) for the squared term, \(b\) for the linear term, and \(c\) for the constant term. By substituting these coefficients, we obtain the solutions for \(x\) that zero the function. In our example with \(g(x)=5x^{2}-10x-5\), we identify \(a=5\), \(b=-10\), and \(c=-5\). Plugging these into the quadratic formula gives us the zeros.
This formula is powerful because it works for any quadratic function, regardless of whether the function factors neatly or not. Hence, the quadratic formula is one of the most reliable methods for finding zeros algebraically.
The quadratic formula is:\[x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\]
It utilises the coefficients of the quadratic equation: \(a\) for the squared term, \(b\) for the linear term, and \(c\) for the constant term. By substituting these coefficients, we obtain the solutions for \(x\) that zero the function. In our example with \(g(x)=5x^{2}-10x-5\), we identify \(a=5\), \(b=-10\), and \(c=-5\). Plugging these into the quadratic formula gives us the zeros.
This formula is powerful because it works for any quadratic function, regardless of whether the function factors neatly or not. Hence, the quadratic formula is one of the most reliable methods for finding zeros algebraically.
Graphing Quadratic Functions
Graphing quadratic functions is not only an effective visual tool but also a strategic approach to understand the properties of quadratics, such as symmetry, vertex, and direction of opening. A quadratic function typically creates a parabola on the graph.
When graphing \(g(x)=5x^{2}-10x-5\), we look for the points where the parabola crosses the x-axis. These points are the graphical representation of the function's zeros. The use of a graphing utility can provide a visual aid and further verify the zeros found algebraically.
By seeing the parabola and its intersection with the x-axis, students can develop an intuitive understanding of how the coefficients of the equation influence the position and shape of the graph. Comparing the algebraic solutions with the graphical solutions can reinforce the concept of zeros and the reliability of both methods. Moreover, graphing can be particularly useful when the algebraic method seems daunting or infeasible.
When graphing \(g(x)=5x^{2}-10x-5\), we look for the points where the parabola crosses the x-axis. These points are the graphical representation of the function's zeros. The use of a graphing utility can provide a visual aid and further verify the zeros found algebraically.
By seeing the parabola and its intersection with the x-axis, students can develop an intuitive understanding of how the coefficients of the equation influence the position and shape of the graph. Comparing the algebraic solutions with the graphical solutions can reinforce the concept of zeros and the reliability of both methods. Moreover, graphing can be particularly useful when the algebraic method seems daunting or infeasible.
Other exercises in this chapter
Problem 38
Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$g(x)=-\frac{x}{(x-2)^{2}}$$
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(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts o
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Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$g(x)=1+\frac{6}{x-3}$$
View solution Problem 39
Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=
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