Algebra II Review - Units 1 & 2

The y-intercept is the point where the graph crosses the y-axis. To find the y-intercept, set x = 0 and solve for y.

The coefficient of the x² term determines the width of the graph. A larger absolute value of the coefficient means a narrower graph.

A quadratic equation in the form ax² + bx + c = 0 can be solved using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. If the discriminant (b² - 4ac) is negative, the solutions are complex numbers.

The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex. To convert from standard form to vertex form, complete the square.

The discriminant of a quadratic function is the expression b²-4ac. It tells us about the nature of the roots (solutions): If the discriminant is positive, there are two distinct real roots. If it is zero, there is one real root. If it is negative, there are two complex roots.

To multiply binomials, use the distributive property or the FOIL method (First, Outer, Inner, Last): (ax + b)(cx + d) = acx² + adx + bcx + bd.

The equation of a circle with center (h, k) and radius r is: (x - h)² + (y - k)² = r².

A polynomial function can be factored into linear factors. To find the zeros, set each factor equal to zero and solve. The multiplicity of a zero is the number of times a factor appears in the factored form. This tells us how the graph behaves near that zero.

The value of x that makes the polynomial equal to zero. A polynomial's roots, also known as solutions, are the values of x at which it crosses the x-axis. Therefore, an x-intercept is a root.

A polynomial is divided by (𝑥 + 1) and the result is expressed as 2𝑥 3 − 3𝑥 2 + 𝑎𝑥 + 𝑏 + 𝑐/𝑥+1. The sum of 𝑎, 𝑏, and 𝑐 is 13.

A polynomial can have multiple roots, and some roots might repeat. This indicates that the graph of the polynomial touches the x-axis at that point but doesn't cross it.

The Factor Theorem states that if a polynomial ℎ(𝑥) has a root of −2, then (𝑥 + 2) is a factor of the polynomial. The Remainder Theorem states that the remainder when ℎ(𝑥) is divided by (𝑥 + 2) is 0 because (𝑥 + 2) is a factor.

A polynomial's end behavior is determined by its leading term's degree and coefficient. If the degree is even, the ends of the graph point in the same direction. If the degree is odd, the ends point in opposite directions.

The polynomial 𝑝(𝑥) = 𝑥 3 − 2𝑥 2 − 23𝑥 + 60 has a factor (𝑥 + 5). This means (𝑥 + 5) divides the polynomial. The other two factors are found by factoring 𝑥 2 − 7𝑥 + 12 = (𝑥 − 3)(𝑥 − 4)

If a polynomial has a real coefficient and complex root, then its conjugate is also a root. Therefore, if 𝑥 = −6 is a real root, then the other root could be the complex conjugate of 𝑥 = −6.

Based on the graph: a) The x-intercepts are (0,0), (-2, 0), and (1, 0), which confirms the factors (x, x+2, and x-1). b) The graph touches the x-axis at (0, 0) and (-2,0), which indicates that the factors are squared. c) The graph does not touch the x-axis at (1, 0), which indicates that the factor (x-1) is not squared. d) The graph has the correct end behavior, as both ends point upwards, which indicates that the leading term is positive and has an even degree.

To solve an absolute value inequality like 2|𝑥 + 3| > 6, follow these steps: 1) Isolate the absolute value expression (2|𝑥 + 3| > 6 becomes |𝑥 + 3| > 3). 2) Set up two inequalities, one with the absolute value expression greater than the positive value on the right and the other less than the negative value (𝑥 + 3 > 3 and 𝑥 + 3 < −3). 3) Solve for 𝑥 in each inequality (𝑥 > 0 and 𝑥 < −6). 4) Express the solution in interval notation, which represents all the possible values of 𝑥 that satisfy the inequality. The solution is (-∞, -6) U (0, ∞).

To solve an absolute value equation like 2|𝑥 − 1| − 7 = 2, follow these steps: 1) Isolate the absolute value expression (2|𝑥 − 1| − 7 = 2 becomes 2|𝑥 − 1| = 9, and then |𝑥 − 1| = 9/2). 2) Set up two equations, one with the absolute value expression equal to the positive value on the right and the other equal to the negative value (𝑥 − 1 = 9/2 and 𝑥 − 1 = −9/2). 3) Solve for 𝑥 in each of these equations, and get 𝑥 = 11/2 and 𝑥 = −7/2. 4) Calculate the sum and product of the solutions (11/2 + (-7/2) = 2 and (11/2)*(-7/2) = -77/4).

To evaluate a function at a given input, substitute the input value for the variable in the function's expression and simplify. To evaluate 𝑓(−3) for 𝑓(𝑥) = −𝑥 3 + 4𝑥 2 − 5𝑥 + 10, substitute −3 for 𝑥 in the function: 𝑓(−3) = −(−3) 3 + 4(−3) 2 − 5(−3) + 10. Simplify the expression: 𝑓(−3) = 27 + 36 + 15 + 10 = 88. Therefore, 𝑓(−3) = 88.

A system of linear equations represents a set of two or more equations with the same variables. To find the product of the solutions to a system of linear equations, first solve the system for the values of the variables that satisfy both equations. This solution point represents the intersection of the lines represented by the equations. Then, multiply the x and y coordinates of the solution point together. For example, if the solution to the system is (2, 3), then its product would be 2 * 3 = 6.

A linear equation in slope-intercept form is expressed as y = mx + b, where m is the slope of the line, and b is the y-intercept. To write the equation of a line in slope-intercept form, you need two pieces of information: 1) the slope (m) and 2) the y-intercept (b). You can use a given point and the slope to find the y-intercept. If you have two points on the line, you can calculate the slope and use one of the points and the slope to find the y-intercept.

A parabola is a U-shaped curve that is the graph of a quadratic function. The axis of symmetry of a parabola is a vertical line that divides the parabola into two symmetrical halves. The equation of the axis of symmetry can be found using the formula x = -b/2a, where a and b are the coefficients of the quadratic function y = ax² + bx + c. If the parabola passes through two x-intercepts (points where the parabola crosses the x-axis), the axis of symmetry will be the vertical line that passes through the midpoint of the x-intercepts.

A quadratic function is a polynomial function where the highest power of the variable is 2. It can be written in the form y = ax² + bx + c, where a, b, and c are constants with a ≠ 0. The graph of a quadratic function is a parabola. The factored form of a quadratic function is expressed as y = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots or x-intercepts of the parabola (where the graph crosses the x-axis). The value of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0).

A quadratic function can be represented in standard form (y = ax² + bx + c) or vertex form (y = a(x - h)² + k). The vertex form provides direct information about the vertex of the parabola (h, k), which is the highest or lowest point on the graph. The vertex form also reveals the axis of symmetry, which is the vertical line x = h. The value of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0).