Number of Local Extrema Graph the polynomial, and determine how many local maxima and minima it

Number of Local Extrema Graph the polynomial, and determine...

Key Concepts

Polynomial Functions

Polynomials are algebraic expressions involving sums of powers of a variable with coefficients. They are central to algebra and calculus, and their behavior depends on the degree of the polynomial. Understanding polynomials is key to analyzing how functions behave over the real numbers, including their growth, curvature, and end behavior.

Quadratic Functions

Quadratic functions are a specific type of polynomial function with degree two, typically expressed in the form ax² + bx + c. They produce parabolic graphs, and the sign of the leading coefficient determines whether the parabola opens upward or downward, which is essential for determining the location of local extrema such as the vertex.

Local Extrema

Local extrema are the points on a function's graph where the function reaches a local maximum or minimum value. These points can be identified through analysis methods such as finding the vertex in quadratic functions or using derivative tests in calculus. They provide insight into the function's behavior near specific points, helping to understand its overall shape.

Graphing Functions

Graphing is the process of plotting a function on a coordinate system to visualize its behavior. Through graphing, one can identify critical features such as intercepts, turning points, symmetry, and local extrema. This practice is crucial in understanding how the function behaves across different intervals and in identifying key characteristics like the maximum or minimum values.

Transcript

00:02 So if we wanted to sketch this polynomial, well, the first thing we have to do is factor to determine the zeros.

00:08 So if we multiply our constant by our coefficient to x squared, what we get is x squared plus 3x minus 10.

00:21 And so we want two numbers that multiply to negative 10, but add up to positive 3.

00:27 So this would be x plus 5 and x minus 2.

00:32 But remember, since we multiplied by negative 2, we have to now divide by negative 2.

00:37 So this would be negative 2, and this would be negative 2.

00:40 So we can rewrite this as negative 2x plus 5, and then x plus 1.

00:49 So now we know our 0s.

00:51 Our 0s would be, let's if we say negative 2x plus 5 is equal to 0, we would have negative 5 over negative 2.

00:59 So this gives x is equal to 5 over 2, and of course x is equal to negative 1.

01:03 And both of these have a multiplicity of 1.

01:07 So now let's look at our boundary.

01:09 So as x approaches infinity, our polynomial would approach, would approach positive, or rather, this would be negative infinity times positive infinity.

01:19 So we would approach negative infinity.

01:22 And as x approaches, as x approaches negative infinity, well our polynomial here would be this negative, this positive infinity times a negative infinity.

01:31 So again, we're going to be negative infinity...

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