Absolute Value Transformations and Equations

Front

What is the definition of absolute value?

Back

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x|.

Front

How do you solve an equation involving absolute value, such as |x + a| = b?

Back

To solve |x + a| = b, you set up two separate equations: x + a = b and x + a = -b, then solve for x in both cases.

Front

What are the solutions to the equation |x - 5| = 3?

Back

The solutions are x = 8 and x = 2.

Front

What does it mean for a function to have a minimum or maximum value?

Back

A minimum value is the lowest point on the graph of a function, while a maximum value is the highest point.

Front

How do you determine the minimum or maximum of a quadratic function?

Back

For a quadratic function in the form f(x) = ax^2 + bx + c, the vertex formula x = -b/(2a) gives the x-coordinate of the vertex, which is the minimum or maximum point.

Front

What is the vertex of the function f(x) = |x - 3| + 2?

Back

The vertex is at the point (3, 2).

Front

How do you graph an absolute value function?

Back

To graph an absolute value function, identify the vertex, determine the direction of the opening (upward or downward), and plot points on either side of the vertex.

Front

What is the effect of a negative sign in front of an absolute value function, such as -|x|?

Back

A negative sign in front of an absolute value function reflects the graph across the x-axis.

Front

How do you solve the equation |2 - x| = 6?

Back

Set up two equations: 2 - x = 6 and 2 - x = -6. Solving these gives x = -4 and x = 8.

Front

What is the solution to the equation |x + 12| = 0?

Back

The solution is x = -12, since the absolute value is zero only when the expression inside is zero.