Matching In Exercises 7-12, match the quadratic function with its graph. [The graphs are labeled

Matching In Exercises 7-12, match the quadratic function...

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that are generally expressed in the form f(x) = ax² + bx + c. Their graphs are parabolic shapes which can open upward or downward depending on the sign of the coefficient a. They are fundamental in algebra and are used to model various real-world phenomena due to their predictable symmetry and vertex characteristics.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it easier to identify the location of the vertex and understand how the graph of the quadratic function is shifted from the basic quadratic function f(x) = x².

Graph Transformations

Graph transformations involve shifting, reflecting, stretching, or compressing the graph of a function. In the context of quadratic functions, a horizontal shift is indicated by the (x - h) term and a vertical shift by the constant k. These transformations allow one to predict and sketch the graph of a quadratic function efficiently.

Reflection Transformation

A reflection transformation occurs when the quadratic function has a negative leading coefficient, which reflects the graph across the x-axis. This transformation changes the direction in which the parabola opens: if the original parabola opens upward, a reflection will result in a downward-opening parabola.

Transcript

00:01 Today we're going to be plotting this function on a coordinate grid.

00:05 If this was given in a textbook, this would be matching.

00:09 But even though we don't have the given graphs, we still can be able to do it on our own.

00:15 We're going to start out by distributing the negative sign into the parentheses.

00:22 So we are going to do negative times, and i'll write ffx, which is just our.

00:30 Function or a y and a negative sign times x is negative x and then negative times negative four is going to be positive four and all of this is still going to be squared and the fact that our function is being squared tells us that it's going to be a parabola which looks like a u and the fact that we have to distribute a negative sign means that our parabola is going to be upside down.

01:05 And so now we need to find our x intercept.

01:10 And the way that we do this is by setting our y equal to zero.

01:18 And we can do this two ways.

01:20 And i will discuss both ways.

01:22 So in the first way, we are going to focus just on this right side of our function, because we already know the left side is equaling 0.

01:31 And we don't have to focus on the squared part because we know once we find what x will equal here, we will be good.

01:38 And so i will just copy the negative x plus 4 and i set it equal to zero because again, i'm trying to find my x intersect.

01:49 And so i would subtract the constant.

01:53 I would subtract 4 on both sides and that will leave me with negative 4 equals.

02:01 Negative x and then i need to divide both sides by negative sign or you can give it as a negative one and so this tells me that x equals four and so one part is looking at our right side of our function and then the other way of finding the external step is by using the quadratic formula and so to find my quadratic formula, i need to first write this as a quadratic equation.

02:43 So i need to do negative x plus 4 times negative x plus 4 because it's being squared.

02:56 And 4 times 4 is 16.

03:00 4 times negative x is negative 4x.

03:05 Negative x times four is negative 4x and negative x times negative x is x squared and make sure when you're doing this that you line everything up correctly make sure that you're laying up your terms with their like terms because those are the only things that we can combine and i'm going to bring down my expression so i have 16 at positive 16 negative 4x plus negative 4x is negative 8x, and then i have x squared.

03:43 And so that is my quadratic equation.

03:46 And so i'm going to substitute these values into the quadratic formula.

03:53 And the quadratic formula is negative b plus or minus the square root of b squared minus 4ac, and all of that is over 2a.

04:13 And now i'm going to start substituting from the blue quadratic equation that we just made.

04:20 And the way that we know what our a, b, and c, our a, is the coefficient that is front of our x squared, which even though there is no number there, it is one.

04:31 There is an understood one.

04:33 Anytime there is a variable by itself.

04:35 My b is in front of the x.

04:39 And the 16 is the c.

04:42 It is our constant.

04:44 And so now i'm going to set the two.

04:46 So i have negative b times negative 8, because that's replacing the b, plus or minus the square root of b squared.

05:00 So that's negative 8 squared minus 4 times 1 times 16.

05:16 And all of that is going to be over 2 times 1.

05:22 1 because that is my 2a.

05:26 And then i'm just going to continue through this.

05:29 Negative times negative 8 is positive 8 plus or minus the square root.

05:37 Negative 8 square is 64.

05:41 Negative 4 times 1 is negative 4 and negative 4 times 16 is negative 64.

05:49 And 2 times 1 is 2.

05:55 And i have 8 plus or minus the square root of 64 minus 64 to 0.

06:06 So 8 plus or minus the square root of 0.

06:12 And all that is going to be divided by 2.

06:16 And then i scroll down a little.

06:18 And then i have 8 plus or minus the square root of 0 is just 0.

06:24 And all that is over 2.

06:26 And 8 plus 0, 8 minus 0 is 8, and 8 divided by 2 equals 4.

06:35 So as you can see, either way, we see that our x intercept, the place where what x will equal on our corner grid when y equals 0 is going to be 4, whether you use the quadratic formula or not.

06:51 And so now i'm going to start filling in my values table.

06:55 So i can actually start working toward being able to plot this.

06:59 So i have my x values, and then i'll have my y values.

07:06 And so we already found out that when y is zero, x will equal four.

07:14 And so now we're going to be able to use that original equation that was given.

07:18 So the original equation, again, was f of x equals negative x minus 4 squared.

07:32 And we're going to be able to use that equation, and we're going to start looking at numbers that are immediately around the four, but only on one side, to be able to start working through this.

07:42 And we only need to focus on one side of the four, either the numbers that are less than four or the numbers that are greater than four because we're going to use symmetry to do the other side...

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