Science:Math Exam Resources/Courses/MATH110/December 2015/Question 03 (b)

MATH110 December 2015

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Question 03 (b)

Consider the following function, $f(x)={\begin{cases}e^{x}&x\leq 0\\ax+b&0<x<1\\-x^{2}+x&x\geq 1.\end{cases}}$ where $a$ and $b$ are constants. (b) With the values of $a$ and $b$ found in part (a), sketch the graph of the corresponding function $f$. Make sure you evaluate and mark on your graph the $x$− and $y$−intercepts, if they exist. (If you didn’t answer part (a), pick values of a and b of your choice and sketch the graph of the corresponding function for partial marks.)

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Sketch the graph of each piece separately, i.e. an exponential function, a line and a parabola. For each graph keep the piece that satisfies the given $x$ interval.

Checking a solution serves two purposes: helping you if, after having used the hint, 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

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'd like to graph the function: $f(x)={\begin{cases}e^{x}&x\leq 0\\-x+1&0<x<1\\-x^{2}+x&x\geq 1.\end{cases}}$ The graph is shown in the figure below. The red part is the function $e^{x}$ for $x\leq 0$, The Green part is the linear function $-x+1$ for $0<x<1$, and the orange part is the parabola $-x^{2}+x$ where $x\geq 1$. As shown in the graph, the $x$-intercept is $x=1$, and $y$-intercept is $y=1$ .