1.2: Homework for Algebra Tips and Tricks- Part I
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Simplify the following algebraic expressions.
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\((y + 2)^2 - (y-1)^2\)
\(6y + 3\)ans
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\(\frac{5x^3 + 6x^2 + x}{x}\)
\(5x^2 + 6x + 1\)ans
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\(\frac{(x-3)^2 - 9}{x}\)
\(x-6\)ans
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\((y + 2)^2 - (y-1)^2\)
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Simplify the following.
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\((x - 3)(x + 1)(x - 2)\)
\(x^3 - 4x^2 + x + 6\)ans
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\((x + 1)^3\)
\(x^3 + 3x^2 + 3x + 1\)ans
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\((x + h)^3\)
\(x^3 + 3x^2h + 3xh^2 + h^3\)ans
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\((x-1)^3 - x^3 - 1\)
\(-3x^2 + 3x - 2\)ans
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\((x - 3)(x + 1)(x - 2)\)
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Given the functions \(f(x) = 5x - 10\) and \(g(x) = 3x + 4\), find the following.
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\(f(7)\)
\(25\)ans
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\(g(4)\)
\(16\)ans
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\(f(x + 3)\)
\(5x + 5\)ans
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\(g(3y - 2)\)
\(9y - 2\)ans
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\(f(g(x))\)
\(15x + 10\)ans
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\(f(7)\)
- Sketch graphs of \(e^x\), \(\frac{1}{x}\), and \(\sin(x)\). (if you don’t know what it looks like, use a calculator or look up the answer on the internet).
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For the following graph, imagine that it represents the amount of money in Rebecca’s bank account. Create a story that explains the various ups and downs.
Answers vary. Here is one possibility (please don’t use this, come up with your own!): Rebecca is a money counterfeiter. Business is booming in the 1990s, and she makes (literally) a boat load of cash, launders it, and makes bank. However, the Feds in the 2000s came out with these new benjamins with watermarks and stuff like that, and she can’t counterfeit it anymore. She loses all her money gambling on water polo. She tries to get her business back a couple times, but it never catches on. Finally, she decides to invest in bitcoin when it was trading at $1 USD per bitcoin, and then she made a serious fortune.ans
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\(0^\circ\) in Celsius is \(32^\circ\) in Fahrenheit, and \(100^\circ\) in Celsius is \(212^\circ\) in Fahrenheit.
- Sketch a graph with Fahrenheit along the \(x\)-axis and Celsius along the \(y\)-axis. Hint: I’d start with the points \((32, 0)\) and \((212, 100)\), and connect them with a straight line.
- What is the slope of the graph from part (a)?
- What is a formula to convert from Fahrenheit to Celsius?
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For each graph below, state whether it is a function or not. (This involves using the “Vertical Line Test”)
A and C are functions, B and D are not.ans
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Let \(f(x) = 15 - 2x^2\).
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Find the slope of a line that goes through the points \((-1, 13)\) and \((0, 15)\), both of which lie on the graph of \(f(x)\).
Slope is \(2\)ans
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Consider the function \(f(x) = 15 - 2x^2\). Find the slope of a line that intersects this curve at \(x = 1\) and \(x = 2\).
Slope is \(-6\)ans
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Find the slope of a line that goes through the points \((-1, 13)\) and \((0, 15)\), both of which lie on the graph of \(f(x)\).