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What rules must be followed when plotting the graph of an inequality with an equal sign?
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Use a solid line for the boundary and shade the region that satisfies the inequality.
What are the domain and range of quadratic functions?
The domain of quadratic functions is all real numbers, while the range depends on the direction of the parabola (U-shaped graph).
What is the primary goal when working with equations in algebra?
To isolate the variable and solve for its value.
Explain the concept of a piecewise function.
A piecewise function is defined by multiple sub-functions, each applying to a certain interval of the main function's domain.
What are the key characteristics of continuous functions?
Continuous functions have no holes or gaps in their domain.
What are the different forms of linear equations?
Slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), and standard form (Ax + By = C).
What is the quadratic formula and when is it used?
The quadratic formula is x = [-b ± √(b² - 4ac)] / 2a and it is used to find the solutions of a quadratic equation.
Explain the vertical line test and its purpose.
The vertical line test is used to determine if a graph represents a function. If a vertical line intersects the graph at more than one point, it’s not a function.
What are the rules for exponents that are critical to remember for exponential functions?
Key rules include the product rule (a^m * a^n = a^(m+n)), quotient rule (a^m / a^n = a^(m-n)), and power rule ((a^m)^n = a^(m*n)).
Describe the conditions for graphing linear inequalities with no equal sign.
Use a dashed line and shade the area according to the inequality sign (> or <).
What is the significance of the slope in a linear function?
The slope determines the steepness and direction of the line represented by the linear function.
In the context of systems of equations, what are the substitution and elimination methods?
Substitution involves replacing one variable with an equivalent expression from another equation, while elimination involves combining equations to cancel out one of the variables.
Which algebraic rule can be remembered by the acronym PEMDAS?
The order of operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
In the context of functions, differentiate between the independent and dependent variables.
The independent variable (x) is the variable you control, while the dependent variable (y) depends on the value of x.
Define the discriminant in a quadratic equation and its purpose.
The discriminant (b² - 4ac) determines the number of solutions a quadratic equation has. Positive means two solutions, zero means one solution, and negative means no real solutions.
How does the value of the discriminant affect the solutions of a quadratic equation?
A positive discriminant means two distinct real solutions, zero means one real solution, and a negative discriminant means no real solutions.
What does an asymptote represent in an exponential function?
An asymptote is a line that the graph of an exponential function approaches but usually never touches.
How do you determine the x-intercept and y-intercept of a function?
The x-intercept is where the function intersects the x-axis, and the y-intercept is where the function intersects the y-axis.
Describe the three possible forms of quadratic functions.
The three forms are vertex form, standard form, and factored form.
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