Find the vertex of the graph of each quadratic function. Determine whether the graph opens upwar

Find the vertex of the graph of each quadratic function....

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial expressed in the standard form f(x) = ax² + bx + c, where a, b, and c are constants and a is non-zero. This type of function produces a graph in the shape of a parabola, and its properties such as the vertex, axis of symmetry, intercepts, and the direction in which it opens depend on the values of a, b, and c.

Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the parabola reaches its maximum or minimum value. It can be found using the formula x = -b/(2a) for the x-coordinate, and substituting that back into the function to obtain the y-coordinate. This point indicates the highest or lowest point of the graph depending on whether the parabola opens downward or upward.

Parabola Opening Direction

The direction in which a parabola opens is determined by the leading coefficient a in the quadratic function. If a is positive, the parabola opens upward, indicating a minimum vertex. If a is negative, it opens downward, indicating a maximum vertex. This property is crucial for understanding the overall shape and behavior of the graph.

Intercepts

Intercepts are points where the graph intersects the coordinate axes. The y-intercept is found by evaluating the function at x = 0, while x-intercepts are determined by setting the function equal to zero and solving the resulting quadratic equation. These points provide essential information about the graph's location and behavior relative to the axes.

Graph Sketching of Quadratic Functions

Graph sketching involves plotting the key features of a quadratic function, such as the vertex, axis of symmetry, intercepts, and a few additional points for accuracy. Visualizing these elements helps in forming a complete picture of the parabola's shape and its overall behavior, forming an essential part of the analysis of quadratic functions.

Transcript

00:01 Okay, so here we have the function, f of x is equal to x squared minus 2x minus 15.

00:05 And we notice here that we have a quadratic in standard form.

00:10 In the form, f of x is equal to ax squared plus bx plus c, right? well, a is 1, b, well, is a negative 2, because negative 2x is plus a negative 2x.

00:24 And then c, well, negative 15, minus 15 is plus a negative 15.

00:28 So c is minus 15.

00:31 So we know that our vertex of the parabola is going to be at the point negative b over 2a, comma, f of negative b over 2a.

00:47 Okay.

00:48 Let's just go ahead and find, well, negative b over 2a.

00:50 So we have, well, b is a negative 2.

00:54 So we have negative and negative 2.

00:57 That's just positive 2 over 2a or over 2 times 1.

01:03 Well, this is just equal to 2, right? this is minus negative 2 is just 2 over 2 times 1, which is 2.

01:10 Therefore 2 over 2 is equal to 1.

01:13 So therefore our x value of our vertex is just 1.

01:18 Okay? to find the corresponding y value, we just substitute in 1 because we're going to find f of, negative b over 2a.

01:29 Well negative b over 2a we just found to be one.

01:32 So f of negative b over 2a is just f of 1.

01:36 Well f of 1 means just substituting in 1 for x in your function.

01:42 So we have 1 squared minus 2 times 1 minus 15.

01:49 Okay.

01:49 Well that equals 1 minus 2 minus 15 is minus 16.

01:56 So f of 1 is equal to negative 16.

02:00 So therefore, that is our y value of our vertex.

02:03 So therefore, our vertex is at the point 1 comma negative 16.

02:09 Okay, so there we have our vertex.

02:11 Now, to find the x intercepts, while the x intercepts happen where the function is equal to zero.

02:19 Or where y, right, f of x is just a little name for y, really.

02:22 So when y is equal to zero, right, whatever the value of x is there, that's where we're going to intersect the x -axis.

02:31 So to find the x intercepts, you set your function equal to zero, and we have zero is equal to x squared minus 2x minus 15.

02:42 Okay.

02:42 Now, well, you can always use the quadratic equation here to solve for x, or you could just, i mean, i guess an easier way would be if you can factor, always try and factor.

02:54 And this should factor actually pretty nice.

02:56 If you try to factor this, we have zero is equal to.

02:58 If we factor x squared minus 2x minus 15, we get what? we get well, x squared has been x times x.

03:05 Now 15, well, that's three times five, and the difference of five and three is two.

03:10 So i think we're going to need a three and a five, right? i need different signs to multiply to a negative 15.

03:16 I have to add to a negative two.

03:18 So the negative better be on the larger value or the five.

03:23 So we have x plus three times.

03:26 X minus five okay that's factored now so therefore i have two factors two things being multiplied together that's equal to zero so we know that at least one of the factors must be zero so either x plus three is zero well that means that x is equal to negative three right or x minus five is zero which means that x is equal to five okay so therefore our x intercepts are going to be, well, when x is negative three, y is zero.

04:03 So we have the x intercept of negative three, zero...

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