Precalculus

• The equation of the line passing through (x1, y1) with slope m is given by y - y1 = m(x - x1).
• To solve a system of linear equations graphically, plot the graphs of both equations on the same coordinate plane and find their intersection point.
• A system of two or more linear equations can have one solution, no solutions, or infinitely many solutions.
• The final review test is intended to help students study for the upcoming test and is a good place to start studying but it is not the only studying that should be done.
• The test questions may not look the same as the questions below.
• The formatting of the questions, especially the questions on graphs, may be drastically different.
• The equation 𝑥 − 𝑦 + 2𝑧 = 0 can be rewritten as 2𝑥 − 4𝑦 + 5𝑧 = −5.
• The equation 𝑦 − 𝑧 = 2 can be rewritten as 3𝑦 + �� = 2.
• The equation 𝑥 + 2𝑦 = 3 can be rewritten as 3𝑥 − 2�� = −2.
• The equations to be sketched include (𝑥 − 2)2 + (𝑦 + 1)2 = 64, 𝑦 = 𝑥2 − 𝑥 − 12, 𝑦 = 23𝑥 − 5, 𝑔 (𝑥) = 𝑥3 + 6𝑥2 + 9𝑥, �� (𝑥) = − log 2 (2 − 𝑥), 𝑓 (𝑥) = 3𝑥 − 1 − 2, 𝑝 (𝑥) = 2𝑥3 − 5𝑥2 − 4𝑥 + 3.
• The equation or inequality to be solved includes 𝑥1/2 − 3𝑥1/4 + 2 = 0, |𝑥 − 4| < 3, 𝑥 (𝑥 − 1) (𝑥 + 2) ≥ 0, 𝑥�� + 2 − 3 = 1𝑥 + 2, �� ≤ 6 − 𝑥2 �� − 5, log 2 (𝑥 + 2) + log 2 (𝑥 − 1) = 2, 10𝑥 + 3 = 62𝑥, √𝑎 − 12 + √𝑎 = 6, 6𝑥 + 1 = 216, 𝑥2 − 5𝑥 = 12, cot (��) + 1 = 0, tan 𝜃 − 3 cot 𝜃 = 0.
• Barry orders one coffee and three doughnuts and pays $3.75.
• Cathy orders three coffees, one juice, and four doughnuts and pays $9.25.
• The price of coffee, juice, and doughnuts at this coffee shop is $1.50 for a coffee, $2.50 for a juice, and $3.50 for a doughnut.
• In 2½ hours an airplane travels 600 km against the wind.
• It takes 50 min to travel 300km with the wind.
• The speed of the wind and the speed of the airplane in still air are 10 mph and 150 mph respectively.
• The polar curve with equation 𝑟 = 1 + sin 𝜃 is represented as a circle.
• The polar curve with equation 𝑟 = cos 3 𝜃 is represented as an ellipse.
• The polar curve with equation 𝑟 = 3 + cos 3 𝜃 is represented as a hyperbola.
• The polar curve with equation 𝑟 = 2 cos (𝜃 2) is represented as a hyperbola.
• For each complex number, graph the number on a complex plane, write the number in polar form, and find the cube roots.
• The partial fraction decomposition of 3 𝑥 + 1 𝑥 2 − 2 𝑥 − 15 is 4 �� − 1 (�� − 1) 2 (𝑥 + 2).
• The partial fraction decomposition of 4 𝑥 − 1 (𝑥 − 1) 2 (𝑥 + 2) is �� − 1.
• The system {3 𝑥 − 𝑦 = 5, 2 𝑥 + 𝑦 = 5} can be solved or graphed.
• The system {4 𝑥 − 14 𝑦 = 56, 𝑦 = 2 7 𝑥 − 4} can be solved or graphed.
• The system {𝑥 2 + 𝑦 ≤ 5, 𝑦 < 2 𝑥 + 5} can be solved or graphed.
• The system {𝑥 − 2 𝑦 + 𝑧 = 8, 4 𝑥 + �� = 9 − 2 𝑥 + 𝑦 − 𝑧 = − 8} can be solved or graphed using Gauss - Jordan elimination.
• The function (3 - 2𝑖) (4 + 3𝑖) is an example of a rational function.
• The function 1𝑥 - 12𝑥 - 2 is an example of a rational function.
• The function (𝑟2𝑠4 / 3𝑟1/3𝑠)6 is an example of a rational function.
• The function 2𝑎2 - 7𝑎 - 4 3𝑎2 - 13𝑎 + 4 ÷ 𝑎2 - 2𝑎 - 35 3𝑎2 + 14𝑎 - 5 is an example of a rational function.
• The function √75 + 7√3 - √27 is an example of a rational function.
• The function 92 + √7 - is an example of a rational function.
• The function (5 - 𝑖) (6 + 3𝑖) is an example of a rational function.
• The function 2 (3𝑐 + 1) - (𝑐 - 9) - 2 is an example of a rational function.
• The function 2𝑥3 + 3𝑥2 - 19𝑥 + 10𝑥 - 3 is an example of a rational function.
• The function √53 34) is an example of a rational function.
• The function 𝑓 (𝑥) = 𝑥 - 3 2𝑥 + 5 is an example of a rational function.
• The function g (𝑥) = √2 𝑥 + 3 is an example of a rational function.