Science:Math Exam Resources/Courses/MATH110/December 2013/Question 01 (a)

MATH110 December 2013

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Question 01 (a)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample. a) The graph of $f(x)={\frac {1}{2}}((x+3)^{2}-2)$ crosses the x-axis.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
There are multiple ways to see this and each hint corresponds to each solution. For a hint to the first solution, you could always just try to solve this directly using the quadratic formula.

Hint 2
You could also try to solve this be exploiting the fact that the parabola is given in vertex form.

Hint 3
One could also reason by starting with the parabola $\displaystyle y=x^{2}$ and from here trying to get the given parabola by a series of transformations. Argue how these transformations affect the number of roots of the parabola.

Hint 4
One could also use the Intermediate Value Theorem (IVT) to solve this problem.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Note: The first three parts of this question was identical to that of the October Midterm. True: There are multiple ways to see this. Solve the quadratic equation. The function expands to: $f(x)={\frac {1}{2}}x^{2}+3x+{\frac {7}{2}}$. Which then gives: ${\begin{aligned}x&={\frac {-3\pm {\sqrt {9-4\left({\frac {1}{2}}\right)\left({\frac {7}{2}}\right)}}}{2\left({\frac {1}{2}}\right)}}\\&={\frac {-3\pm {\sqrt {9-7}}}{1}}\\x&=-3\pm {\sqrt {2}}\\\end{aligned}}$

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We can also use the vertex formula by seeing that $f(x)={\frac {1}{2}}(x+3)^{2}-1$, with vertex $(-3,-1)$ which is below the x-axis. Next, observe that a parabola is continuous (because its a polynomial). Combining this with the fact that the parabola has a positive coefficient of ${\frac {1}{2}}$, meaning that the parabola is curving upwards, we get that it must cross the $x$ axis.

Solution 3

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We can view this as a graph transformation of $y=x^{2}$ which touches the $x$-axis. Then we shift it $3$ to the left and then $2$ down, so it still touches the $x$-axis. This is then followed by a vertical compression of $2$ (or expansion by $0.5$). This does not change direction of curvature of the function and so it must still touch the $x$-axis.

Solution 4
Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We can also use the IVT. The function is a quadratic and hence is continuous. We then look at $f(-3)=-1<0$ and then $f(0)=3.5>0$. Thus there must be a point between $-3$ and $0$ that crosses the $x$-axis.

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