(a) find the vertex, the axis of symmetry, and the maximum or minimum function value and (b) gra

step 1 All right, this is the vertex form of a quadratic. And we want to take our function that we're g

(a) find the vertex, the axis of symmetry, and the maximum...

Key Concepts

Quadratic Function

A quadratic function is a polynomial of degree two and is generally expressed in the standard form f(x) = ax² + bx + c. It describes a parabolic curve which either opens upward if the coefficient a is positive or downward if a is negative. Understanding the standard form is essential as it provides the basis for determining key properties such as the vertex, axis of symmetry, and the function's maximum or minimum value.

Vertex

The vertex of a parabola is the point where the function attains its maximum or minimum value, depending on whether the parabola opens downward or upward. It is a critical feature in the analysis of quadratic functions. The coordinates of the vertex can be determined using the formula x = -b/(2a) for the x-coordinate and substituting back into the equation to find the corresponding y-coordinate.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of a parabola, dividing it into two mirror-image halves. In quadratic functions expressed in standard form, the axis of symmetry can be derived directly using the formula x = -b/(2a). This concept is fundamental in understanding the geometric properties of parabolas and simplifying their analysis.

Maximum or Minimum Value

Depending on the direction in which the parabola opens, a quadratic function will have either a maximum or a minimum value, which is the y-coordinate of the vertex. When the coefficient a is negative, the parabola opens downward and the vertex represents the maximum value; when a is positive, the parabola opens upward and the vertex represents the minimum value. This determination is key in optimization problems involving quadratic functions.

Graphing Parabolas

Graphing a quadratic function involves plotting the vertex, axis of symmetry, and additional points that illustrate the curvature of the parabola. Understanding how to locate and mark the vertex and symmetry axis assists in accurately sketching the graph. This process not only visualizes the function's behavior but also reinforces the understanding of quadratic properties and transformations.

Transcript

00:01 All right, this is the vertex form of a quadratic.

00:06 And we want to take our function that we're given, which is f of x equals negative 3x squared minus 7 x plus 2.

00:28 And we want to get it into this form in order to do this.

00:31 Let's think of it as an equation.

00:34 We want to get it into this form, by the way, so that we can pull information out of it easily, like the vertex axis of symmetry, so on and so forth.

00:45 So the next thing we want to do is move the two over.

00:48 So we're going to subtract two to both sides.

00:50 And we want to, just like in this model here, we want to factor out a coefficient, a leading coefficient.

00:58 So we're going to factor out the negative three.

01:00 That's going to leave us with x squared plus seven thirds x.

01:10 I'm going to leave a spot to complete the square.

01:12 We're going to need to use that method in order to our.

01:14 Achieve our goal here.

01:18 All right.

01:19 Now we've got negative 2 equals negative 3 times x plus something square.

01:33 We wanted in a perfect square binomial.

01:35 To determine what that number is, we take half of the coefficient on x.