Solving Linear and Quadratic Systems - Graphically

Front

What is a linear function?

Back

A linear function is a function that can be graphically represented as a straight line. It has the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What is a quadratic function?

Back

A quadratic function is a polynomial function of degree 2, typically in the form y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0.

Front

How can you determine the number of solutions in a linear-quadratic system?

Back

The number of solutions can be determined by analyzing the intersection points of the linear and quadratic graphs. If they intersect at one point, there is one solution; if they intersect at two points, there are two solutions; and if they do not intersect, there are no solutions.

Front

What does it mean for a system of equations to have one solution?

Back

A system of equations has one solution when the graphs of the equations intersect at exactly one point.

Front

What is the significance of the vertex in a quadratic function?

Back

The vertex of a quadratic function is the highest or lowest point on the graph, depending on the direction of the parabola. It represents the maximum or minimum value of the function.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is y = ax^2 + bx + c, where a, b, and c are constants.

Front

What is the vertex form of a quadratic function?

Back

The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.

Front

How do you find the x-intercepts of a quadratic function?

Back

To find the x-intercepts of a quadratic function, set y = 0 and solve the equation ax^2 + bx + c = 0 using factoring, completing the square, or the quadratic formula.

Front

What is the quadratic formula?

Back

The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), used to find the solutions (x-intercepts) of a quadratic equation.

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is the part of the quadratic formula under the square root, b^2 - 4ac. It determines the nature of the roots: if it's positive, there are two real solutions; if zero, one real solution; if negative, no real solutions.