In Problems 59-70, (a) graph each function. (b) Determine...

In Problems $59-70,$ (a) graph each function. (b) Determine the domain and the range of the function. (c) Determine where the function is increasing and where it is decreasing.
$$
f(x)=2 x^{2}+x+1
$$

Key Concepts

Increasing and Decreasing Intervals

Identifying increasing and decreasing intervals of a function involves determining the intervals over which the function’s output values rise or fall as the input increases. For quadratic functions, this is often done by examining the vertex; the function is increasing on one side of the vertex and decreasing on the other, depending on whether the parabola opens upward or downward.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values. For quadratic functions, the domain is typically all real numbers. The range is determined by the vertex and the direction in which the parabola opens, providing either a minimum or maximum value that the function can output.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c. They are characterized by a parabolic graph, which opens upward if the coefficient a is positive and downward if a is negative. The vertex of the parabola represents the function’s maximum or minimum value.

Graphing Functions

Graphing functions involves representing the function visually by plotting its points on the coordinate plane. In the context of quadratic functions, this includes determining and plotting the vertex, axis of symmetry, and any intercepts, then sketching the parabolic curve accurately.

Transcript

00:01 So we have f of x is equal to 2x squared plus x plus 1.

00:07 And i don't see how to factor that.

00:10 And we want to graph.

00:14 We want to find part b.

00:16 We want to find what the domain and range is.

00:19 And then part c we want to know where does it increase, where does it decrease.

00:23 So to find the domain, we already medically know what the domain is.

00:27 The domain, we can plug anything we want in and we can get out anything.

00:32 So we can plug anything we want.

00:34 The range, on the other hand, we're going to have to figure out what the vertex is.

00:38 So we know that the graph opens upward and we want to find the vertex.

00:44 And the vertex is going to take place at negative b over 2a.

00:50 And we can find the zeros as well.

00:52 So why don't we just use quadratic formulas? so negative 1 plus or minus the square root of b squared, b squared, minus 4 times.

01:02 A times c and notice how that radical the discriminant and then over to a and that's going to be four so we know that this is negative it's going to end up being negative 7 therefore we won't have any x intercepts the graph will always stay above the x axis and we're going to have a vertex at negative 1 4th and we'll have to figure out what the corresponding y coordinate is negative 1 4th and something and there are no x intercept x intercepts on and our y intercept is going to take place at one and the graph we know opens upward.

01:42 So let's figure out what this vertex is and then we'll know how to describe our range and where it increases and decreases...

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