12.10.24 Formulating Systems of Linear and Quadratic functions

Front

What is a linear equation?

Back

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

What is a quadratic equation?

Back

A quadratic equation is a polynomial equation of degree two, typically written in the form y = ax² + bx + c, where a, b, and c are constants and a ≠ 0.

Front

How do you find the intersection of a linear and a quadratic function graphically?

Back

To find the intersection, graph both functions on the same coordinate plane and identify the points where the two graphs intersect.

Front

What is the vertex of a quadratic function?

Back

The vertex of a quadratic function in the form y = ax² + bx + c is the point (h, k) where h = -b/(2a) and k is the value of the function at h.

Front

What does the discriminant of a quadratic equation tell us?

Back

The discriminant (D = b² - 4ac) indicates the nature of the roots of the quadratic equation: if D > 0, there are two distinct real roots; if D = 0, there is one real root; if D < 0, there are no real roots.

Front

What is the standard form of a quadratic function?

Back

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

Front

How can you convert a quadratic function from standard form to vertex form?

Back

To convert from standard form y = ax² + bx + c to vertex form y = a(x - h)² + k, complete the square.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves, given by the equation x = h, where h is the x-coordinate of the vertex.

Front

What is the significance of the y-intercept in a linear equation?

Back

The y-intercept is the point where the line crosses the y-axis, represented by the value of b in the equation y = mx + b.

Front

How do you determine if a quadratic function opens upwards or downwards?

Back

A quadratic function opens upwards if the coefficient a > 0 and downwards if a < 0.