Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of

Write the quadratic function in standard form and sketch its...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ? 0. This type of function produces a parabola when graphed, and the shape and position of the parabola provide insight into the behavior of the function across its domain.

Standard Form

The standard form of a quadratic function is written as f(x) = ax² + bx + c. This form is useful because it clearly displays the coefficients that affect the function's shape and position, such as the coefficient a which determines the opening direction and width of the parabola, as well as the constant c which indicates the y-intercept.

Completing the Square

Completing the square is a method used to transform a quadratic function from its standard form into its vertex form. This process involves reorganizing and grouping terms so that the quadratic can be expressed as a perfect square plus a constant, making it easier to identify the vertex and other key features of the parabola.

Vertex

The vertex is the point on the graph of a quadratic function where it attains its maximum or minimum value. In the vertex form f(x) = a(x - h)² + k, the vertex is given by the coordinates (h, k). This point represents the pinnacle of the parabola, and its location is crucial for understanding the function’s overall behavior.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex, and its equation is given by x = h in the vertex form, or alternatively x = -b/(2a) in the standard form. This concept is important for graphing the function and understanding its geometric properties.

X-intercepts

The x-intercepts, or roots, of a quadratic function are the points where the graph crosses the x-axis (i.e., where f(x) = 0). These intercepts can be found using factoring, the quadratic formula, or completing the square. They provide valuable information about the solutions to the quadratic equation and the characteristics of the graph.

Transcript

00:01 Okay, here we have a problem that has us to take a particular function, f of x, and put it into standard form, which i have written right here.

00:09 M of x is equal to some constant a times x minus h quantity squared plus k.

00:14 The good thing about that function is that it gives us a vertex very easily as hk.

00:20 Now, what we want to do is look at these first two terms, all the terms involving x, and see if we can get this entire expression here to look like a, perfect square.

00:33 Now the perfect square that begins x squared minus 8x when expanded out is x minus four quantity square, all right? but we have a 21 here whereas that expression has a 16.

00:44 So what we need to do is subtract five from both sides.

00:48 So we've got f of x minus five equals x minus four quantity squared.

00:55 Now all we have to do is add that five back over to the other side of this equation and we've got our standard form.

01:06 Now the vertex is going to be four, five, it's just hk, which means our axis of symmetry is going to be at x equals four.

01:16 And our vertexes are the points where f of x equals zero.

01:20 Now we could take any one of these functions and plug in f of x equals zero.

01:26 So i'm going to take this one because if i take this one, i would just have to take the five back to the other side.

01:32 So i'm going to take negative 5 is equal to x minus 4 quantity squared and unsquared this whole thing...

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