Find each of the following to the nearest integer by referring to the graphs for Problems 17 and

step 1 Okay, so let's talk about quadratic functions. Quadratic functions like most functions have grap

Find each of the following to the nearest integer by...

Key Concepts

Range

The range of a function encompasses all the output values the function can produce. It is determined by the behavior of the graph, including any extreme values and the limits of the function. Knowing the range is essential for understanding the scope and limitations of the function.

Extreme Values (Maximum or Minimum)

Extreme values refer to the highest or lowest point on the graph of a function. In quadratic functions, this extreme value is found at the vertex and indicates the optimal value of the function. It is important in optimization and in understanding the overall behavior of the graph.

Graph Interpretation

Graph interpretation involves analyzing the visual representation of a function to extract important information such as intercepts, vertex, extreme values, and the range. This skill is particularly useful when exact values must be estimated from the graph, such as when rounding to the nearest integer.

Intercepts

Intercepts are the points where the graph of a function crosses the coordinate axes. The x-intercepts occur where the function's output is zero, and the y-intercept occurs when the input is zero. Understanding intercepts helps in sketching the graph and analyzing the behavior of the function with respect to the axes.

Vertex

The vertex is the point on the graph of a quadratic function where it attains its maximum or minimum value. This point serves as the turning point of the graph and is crucial for determining the function’s symmetry and direction (whether it opens upward or downward).

Transcript

00:01 Okay, so let's talk about quadratic functions.

00:04 Quadratic functions like most functions have graphs.

00:08 The graphs are parabolas.

00:10 And any type of graph, we could talk about, some of the characteristics i have here, we could talk about the intercepts, definitely the range of any function, but for sure, if you're talking about quadratic functions, you'll have a vertex and a maximum, or it could be a minimum.

00:27 So let's kind of focus on, you know, just one of these.

00:33 One of these, probably we have one, two, three, four parabolas, but let's just focus on one.

00:40 Say we look at the graph of the one labeled n.

00:48 And so this one is a parabola.

00:51 I would say this opens down.

00:53 And then we'll talk about what that means as far as one of these characteristics.

00:58 But let's look at, let's start by looking at, at the intercepts.

01:02 And so this graph will have x intercepts and y intercepts.

01:07 When it says intercepts, that means where it crosses each axis.

01:11 And so here we have the x axis and we have the y axis.

01:16 So let's start with the x intercepts.

01:19 So the x i and t, the x intercepts.

01:23 Let's see, where does it cross the x axis is at these points right here.

01:33 These x intercepts what are the coordinates for the x intercepts well it crosses at the x axis at 1 0 so 1 comma 0 so the x coordinate is 1 the y value is 0 and then the other one is at 3 let's not very good 3 let's do 3 0 notice something about the x intercepts well first of all there there are two of them and that can happen with a parabola either has 2 1 or 0 in this case 2 it crosses the x -axis twice.

02:10 But the y value, and this is true for any function, the x -intercepts have a y value of zero or an output of zero.

02:22 But let's look at the y -intercept.

02:24 We also have a y -intercept.

02:26 Where does the graph n cross the y -axis? that's the, looks like you can't really see it.

02:33 There we go.

02:34 It's the y -axis, and it crosses the y -axis right there.

02:37 So that point right there, the y intercept, y intercept, in this case it's reversed from the x intercept in that the x coordinate is zero, and in this case the y is negative 3.

02:54 So zero negative three is the y intercept of this graph.

02:59 And while you can have two x intercepts for this type of graph, for any graph or function, you can only have one y intercept, otherwise it wouldn't be a function, because you'd have the x value of zero are paired with multiple outputs or y values so anyways there's only ever going to be one y intercept with a function okay so part two the vertex so the vertex is kind of i would say it's where the there's a lot of things about the vertex it's where the axis of symmetry crosses three you can imagine if this little dotted line i just guessed here if you're able to fold this screen right along the axis of symmetry, the points on either side of it would line up with each other.

03:47 But the vertex, the axis of symmetry crosses through the vertex, which is this point right here, which if i, so what are the coordinates of that? i would start at the origin, go one, two is the x value, and one is the y value.

04:03 And so the vertex is the xy 2 comma 1.

04:11 Okay, so does this graph n have a maximum or a minimum? well, i would say that this has, why do i think that this has a maximum is because it increases from left to right and then it decreases.

04:29 So when it changes from increasing to decreasing, that is a maximum.

04:33 It's also, you can think of this is the, i guess, the highest point on the graph, the greatest point.

04:40 If we're going to try to identify the minimum, you're going to have trouble.

04:44 It's going to have to involve infinity...

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