12.4.24 Quadratic Inequalities with Graph

Front

What is a quadratic inequality?

Back

A quadratic inequality is an inequality that involves a quadratic expression, typically in the form of ax² + bx + c <, >, ≤, or ≥ 0.

Front

How do you graph a quadratic inequality?

Back

To graph a quadratic inequality, first graph the corresponding quadratic equation as a boundary line. Then, determine which side of the parabola satisfies the inequality by testing a point.

Front

What does the symbol '≥' indicate in a quadratic inequality?

Back

The symbol '≥' indicates that the values of the quadratic expression are greater than or equal to a certain value, meaning the boundary line is included in the solution set.

Front

What does the symbol '>' indicate in a quadratic inequality?

Back

The symbol '>' indicates that the values of the quadratic expression are strictly greater than a certain value, meaning the boundary line is not included in the solution set.

Front

What is the vertex of a parabola in the context of quadratic inequalities?

Back

The vertex of a parabola is the highest or lowest point on the graph, depending on whether it opens upwards or downwards. It is crucial for determining the direction of the inequality.

Front

How do you determine the direction of a parabola from a quadratic inequality?

Back

The direction of a parabola is determined by the coefficient of the x² term: if it is positive, the parabola opens upwards; if negative, it opens downwards.

Front

What is the significance of the x-intercepts in quadratic inequalities?

Back

The x-intercepts (or roots) of the quadratic equation help to identify the intervals on the x-axis where the quadratic expression is positive or negative.

Front

How do you find the solution set for a quadratic inequality?

Back

To find the solution set, graph the quadratic function, determine the intervals where the inequality holds true, and express these intervals in interval notation.

Front

What is interval notation?

Back

Interval notation is a way of writing subsets of the real number line, using parentheses and brackets to indicate whether endpoints are included or excluded.

Front

What does it mean if a quadratic inequality has no solution?

Back

If a quadratic inequality has no solution, it means that there are no values of x that satisfy the inequality, often indicated by the graph not intersecting the x-axis.