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

step 1 Okay, so let's see what we can learn here. We have several quadratic functions that are graphed

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

Key Concepts

Range

The range of a function is the set of all possible output values it can produce. For quadratic functions, the range is particularly influenced by the vertex and the direction in which the parabola opens. It tells us the upper or lower limits of the function's values and is key to understanding the behavior and constraints of the function.

Maximum/Minimum

This concept refers to the extreme values of a function. For a quadratic function, the vertex provides either the highest value (maximum) or the lowest value (minimum). Identifying the maximum or minimum is crucial for optimization problems and for understanding the range and overall behavior of the function.

Intercepts

Intercepts are the points where a graph crosses the coordinate axes. In the context of functions, the x-intercepts (or zeros) are the points where the output is zero, while the y-intercept is the point where the input is zero. These points give important clues about the behavior of the function and are often used to sketch or analyze the graph.

Vertex

The vertex of a quadratic function is its turning point—the point at which the graph changes direction. It represents either the maximum or minimum value of the function depending on whether the parabola opens downward or upward, respectively. The vertex serves as a critical feature for graphing quadratics and understanding their optimal values.

Transcript

00:01 Okay, so let's see what we can learn here.

00:03 We have several quadratic functions that are graphed and other quadracts because their graphs are parabolas.

00:10 We have a lot of them, a lot of things going on here.

00:13 We want to look at the intercepts, the vertex maximum or minimum and range of one of these graphs.

00:21 Let's look at f, the graph of f.

00:27 So we have this parabola.

00:28 It opens up.

00:29 So that will have an impact on what.

00:34 Whether we have a maximum or minimum.

00:36 Let's start, though, with the intercepts.

00:38 We'll find the x intercepts and the y intercepts, meaning where it crosses the x axis and the y axis.

00:45 So the x intercepts, i &t for intercepts.

00:50 Let's see.

00:51 This graph crosses the x axis at these points right here, this point here, and here.

01:00 And so our x intercepts would be, let's see, read the coordinate.

01:05 Start at the origin, go left 1, so that's x equals negative 1, and the y coordinate would be 0.

01:13 You have negative 1, 0, that is one of the x intercept.

01:17 There's another one.

01:18 That would be 3 to the left, so that would be negative 3, and the y coordinate is 0.

01:26 And this format for the x intercept will always be some x value, comma, zero, because on the x -axis, all the y value is always 0.

01:38 Let's find, speaking of why, let's find the y intercept.

01:41 We have the vertical axis, the y axis here, our y intercept that happens where it crosses or hits the y axis.

01:51 And so the coordinates for this, the y intercept, what is the coordinates? what are the coordinates? zero is the x coordinate because starting at the origin, no left or right.

02:04 You just go up in the positive of y direction three units.

02:09 So 0 .3 is the y intercept.

02:12 Because these are functions, you know, all four of these graphs are, they pass the vertical line test.

02:19 Also, there's only ever going to be one y intercept of a function because if x equals zero is paired with multiple y values, then it would fail the vertical line test and would not be a function.

02:30 That's a different problem for a different day.

02:33 Let's look at the vertex.

02:34 You might also think of this as the turning point.

02:38 And there's a lot of symmetry that happens with the vertex.

02:41 Let's see.

02:42 The vertex would be the point right here, sort of in the middle of this parabola.

02:50 And the coordinates of the vertex, coordinates of the vertex, let's see, we go 2 to the left.

02:57 So that's negative x direction and 1 down, negative y direction.

03:02 So it would be negative 2, negative 1.

03:05 The vertex are given by the coordinates negative 2, negative 1.

03:13 Through the vertex, we have our axis of symmetry, our axis of symmetry.

03:21 Now i didn't draw this graph, you know, amazingly, but if i were to, you know, you could fold the, i guess the screen across the axis of symmetry, these points on either side, they're the same distance away, they would line up...

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