Graphs of Quadratic Functions The graph of a quadratic function f is given. (a) Find the coordin

Graphs of Quadratic Functions The graph of a quadratic...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ? 0. This type of function is characterized by a parabolic graph which either opens upward or downward, depending on the sign of the coefficient a. Its study is fundamental in algebra as it provides insights into various properties and behaviors of polynomial functions.

Vertex of a Parabola

The vertex of a parabola is the point where the graph attains its maximum or minimum value. For a quadratic function in standard form, the vertex can be found using the formula (-b/(2a), f(-b/(2a))). The vertex represents the turning point of the parabola, and understanding its location is crucial for graphing the function and solving optimization problems.

Intercepts

Intercepts refer to the points where the graph of a function crosses the axes. The x-intercepts (or roots) are the values of x for which the function equals zero, and they can be found by solving the equation f(x) = 0. The y-intercept is determined by evaluating the function at x = 0. These intercepts are essential for sketching the graph of the function and understanding its behavior at the boundaries.

Extreme Value (Maximum or Minimum)

The extreme value of a quadratic function is either its maximum or minimum, depending on the direction in which the parabola opens. If the parabola opens downward (a < 0), the vertex gives the maximum value; if it opens upward (a > 0), the vertex gives the minimum value. This concept is key in optimization problems and helps in understanding the range of the function.

Domain and Range

The domain of a function refers to the set of all possible input values (usually x-values) for which the function is defined, which for quadratic functions is always all real numbers. The range is the set of all possible output values (y-values) of the function. For a quadratic function, the range is determined by the vertex and the direction in which the parabola opens, making it a crucial aspect of understanding the function's overall behavior.

Transcript

00:01 Okay, so this question gives us a function and asks us to find certain things about the function.

00:06 So first it's asking for the vertex.

00:08 And the way we do that is we need to complete the square.

00:12 To complete the square here, the first thing we need to do is you need to take minus 1 out because it always has to be a, the coefficient of the x squared always has to be 1.

00:23 So it's minus 6x.

00:25 And we keep the minus 5 out of it for now and we'll put that back in later.

00:31 Okay, so that's negative.

00:33 And then we can complete the square of the, the inside the bracket and we can expand that out later.

00:39 So that's x minus three all squared.

00:47 And then that's going to be negative nine.

00:49 It's always going to be negative.

00:51 It's going to be negative nine here because three squared is nine.

00:55 And that's minus five on the outside.

00:58 Okay.

01:00 So we can take the negative back in now.

01:03 So we're going to have negative x minus three all squared plus nine minus five, which is going to be four.

01:14 Okay, so the way we'd get our vertex from this is that we look at our value of x, a value after x inside the bracket and our value outside of the bracket.

01:26 Our value inside of the bracket, the opposite to that is going to be our x ordinate of our vertex.

01:32 So that's going to be three.

01:37 And outside of our bracket, that's going to be our y argument.

01:43 That's three, come a four.

01:44 And that's our vertex.

01:46 And now it's asking us to find the intercepts.

01:51 And the way that we do that is we subbing values of x and y that are 0.

01:57 So we first subbing x that's going to give us x equals 0, that's going to give us our y intercept.

02:05 So that x equals 0, then our f of 0, i'll do this in blue, sorry, f of 0, it's going to be minus x squared, that's just going to be 0 plus 6x also going to be 0 minus 5.

02:26 So our y intercept is going to be 0, minus 5.

02:34 And that's our y intercept.

02:39 And then to find our x intercept, we need to solve the equation.

02:44 And we can do that by setting it equal to 0.

02:48 So we set our, the easiest way to do this because we've already completed the square.

02:52 Then we can just do that by setting our completed the square to zero.

02:59 So we're going to get four on this side, because it goes over to minus four, and then we swap the signs because of this negative...

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