Mathematics Concepts Review

These flashcards cover essential concepts in linear functions, systems of equations, quadratic functions, polynomials, rational expressions, exponential functions, logarithms, sequences, probability, statistics, and trigonometry.

What is the slope of the line given by the equation y = 3x - 7?

The slope is 3. In the slope-intercept form y = mx + b, the coefficient m represents the slope.

What is the y-intercept of the line given by the equation y = -2x + 5?

The y-intercept is 5. In the slope-intercept form y = mx + b, the constant b represents the y-intercept.

Write the equation of a line with a slope of 4 passing through the point (1, -2).

The equation is y = 4x - 6. Using the point-slope formula y - y1 = m(x - x1), we get y - (-2) = 4(x - 1), which simplifies to y + 2 = 4x - 4, then y = 4x - 6. Exploratory steps involve substituting the slope and point coordinates into the linear model.

What is the slope between the points (2, 3) and (6, 11)?

The slope is 2. Calculated using the slope formula m = \frac{y2 - y1}{x2 - x1}, which yields \frac{11 - 3}{6 - 2} = \frac{8}{4} = 2.

What are the solutions for the system of equations y = 2x + 1 and y = x + 4?

The solution is (3, 7). Since both equal y, set the expressions equal: 2x + 1 = x + 4. Subtracting x and 1 gives x = 3. Substitute x = 3 into y = x + 4 to get y = 7. (x = 3, y = 7).

What is the vertex of the quadratic function y = x^2 - 4x + 1?

The vertex is (2, -3). First, find the x-coordinate using x = -\frac{b}{2a}, so x = -\frac{-4}{2(1)} = 2. Then substitute x = 2 back into the equation to find y = (2)^2 - 4(2) + 1 = 4 - 8 + 1 = -3.

In the equation h = -16t^2 + 32t, when does the ball hit the ground?

The ball hits at t = 2 seconds. Set h = 0: 0 = -16t(t - 2). The solutions are t = 0 (start) and t = 2 (hits ground). Formula used: Factoring out the GCF.

What is the 5th term of the geometric sequence where a_1 = 2 and r = 3?

The 5th term is 162. Using the geometric formula an = a1 \cdot r^{(n-1)}, we calculate a_5 = 2 \cdot 3^{(5-1)} = 2 \cdot 3^4 = 2 \cdot 81 = 162.