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

step 1 Okay, we have this diagram here, has several quadratic functions. I know they're quadratics beca

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

Key Concepts

Extrema (Maximum or Minimum)

The maximum or minimum value of a function, known as the extrema, describes the highest or lowest point on the graph. For quadratic functions, the vertex is the extrema. In the context of graph analysis, identifying whether the function has a maximum or a minimum allows one to understand the function’s behavior, particularly in optimization problems and real-world applications.

Range

The range of a function is the set of all possible output values (y-values) the function can produce. In graphical analysis, the range is derived from the vertical spread of the graph. Understanding the range is crucial for comprehending the extent and limitations of the function’s outputs, and it directly relates to the location of the maximum or minimum values on the graph.

Intercepts

Intercepts are points where a graph crosses the coordinate axes. The x-intercepts are the points where the function equals zero and cross the x-axis, while the y-intercept is the point where the input (x) is zero, and the graph crosses the y-axis. Identifying these points on a graph is essential in understanding the behavior and solutions of the function.

Vertex

The vertex of a curve, particularly in the context of quadratic functions, is the point where the function attains its maximum or minimum value. It represents the turning point of the graph and is located exactly halfway between the x-intercepts when the function is symmetric. Knowledge of the vertex is fundamental in graph analysis, as it gives insight into the overall shape of the function.

Transcript

00:01 Okay, we have this diagram here, has several quadratic functions.

00:05 I know they're quadratics because their graphs are parabolas, u -shaped graphs.

00:10 We have four of them here, but we're going to focus on just one of them.

00:14 We're going to look at the graph of g.

00:19 Graph of g.

00:20 So we'll focus on this parabola right here.

00:23 We're going to find its intercepts, its vertex, maximum, minimum.

00:29 Well, it'll be either, you know, one of those.

00:31 We'll have to decide once we get there and then we'll find the range, which is the, when i make this note right now, range is the set of y values or outputs.

00:44 Y values, put the s at the bottom there.

00:49 So the set of y values, once we get to the range, we will see what the range is.

00:53 So the intercepts, though.

00:55 The intercepts are where the graph crosses the x -axis and the y -axis.

01:00 So why don't we start with the x -axis? and we see that graph g crosses the x -axis two times, so it'll have two x intercepts, those two points right there.

01:13 So let's first find the x -i -n -t for intercept.

01:17 So the first one, well, order doesn't necessarily matter, but i would see that the point, this point right here, the coordinates are 1 -0.

01:30 The coordinates, one of the x intercepts can be written as 1 -0, went in the positive x direction one unit, so one to the right, positive one.

01:39 The y coordinate is zero because there's, well, any, the y value is always equal to zero if we're strictly on the x -axis.

01:48 What's the other x intercept? the other x intercept is that one, two, three, four, five, comma, zero.

01:58 So similar format, you'll also have five, zero as the x intercept.

02:04 Oh, well, there are two x intercepts.

02:08 And that can happen often with the quadratic function, just because of the shape of the graph.

02:16 It doesn't always happen that way.

02:18 But in this case, it definitely does.

02:19 We have two x intercepts, but how many y intercepts do we have? let's see.

02:23 Where does it cross the y -axis? the vertical axis? let's see, that's a vertical axis, and it crosses the y -axis, the coordinates.

02:32 And this time, starting at the origin, the x coordinate is zero.

02:37 The x value of our coordinate is zero.

02:39 Going in the positive y direction, one, two, three, four.

02:44 And so the y intercept, the y intercept, y intercept, y intercept.

02:50 Again, we had zero, comma, four.

02:53 So we have our intercepts of this quadratic function, zero four.

02:59 Now we will only ever have a single y intercept for any function because otherwise it would fail the vertical line test and we couldn't actually use that word function.

03:10 All right, so moving on with function g.

03:15 What is the vertex? so the vertex occurs, i would say, kind of in the middle of the graph.

03:21 So this point down here is the vertex.

03:24 So the vertex occurs at this, sort of the coordinates of that point.

03:30 Starting at the origin, you have one, two, three.

03:33 Y values negative 4.

03:37 So the vertex is given by the coordinates 3, negative 4.

03:44 So there we have our vertex.

03:47 Something you may have learned about before is the axis of symmetry, which actually goes through the vertex.

03:53 You can think if we could kind of fold the screen and along the axis of symmetry, the points on either side, they would line up there the same distance away, or at least they should be if i had drawn this accurately.

04:05 Anyways, that is the vertex of this parabola, this function g.

04:10 How about the maximum or minimum? well, let's narrow that down.

04:14 What does this graph have? does it have a maximum or a minimum? well, that is actually related to the vertex.

04:20 So if you look at where the vertex is down here, you can see that's the lowest point on the graph...

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