Graph each parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=3 x^2

Name: Problem 23, Chapter 11: Beginning and Intermediate Algebra
Uploaded: 2022-02-16T17:25:21-08:00
Duration: 3 min 58 s
Channel: Jessica Kluck
Description: Graph each parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=3 x^2

Graph each parabola. Give the vertex, axis of symmetry,...

Key Concepts

Domain and Range

The domain of a quadratic function is the set of all possible x-values, which is typically all real numbers, since quadratic functions are defined for every real x. The range is the set of all possible y-values and depends on the direction in which the parabola opens. For parabolas opening upward, the range is all y-values greater than or equal to the y-coordinate of the vertex, while for those opening downward, it is all y-values less than or equal to the vertex's y-coordinate.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola and divides the graph into two mirror-image halves. The equation of the axis of symmetry is x = -b/(2a), and it is essential for understanding the balance and graphical structure of quadratic functions.

Parabola

A parabola is the U-shaped graph of a quadratic function. It exhibits symmetry and possesses a single turning point called the vertex. The general characteristics of a parabola, such as its direction of opening and width, are determined by the quadratic function's coefficients, making it easier to analyze and sketch the graph.

Quadratic Function

A quadratic function is a second-degree polynomial function typically expressed in the form f(x) = ax² + bx + c. Its graph is a parabola, and the value of the coefficient 'a' determines whether the parabola opens upward (a > 0) or downward (a < 0). Quadratic functions are foundational in algebra and calculus for studying motion, optimization, and the effects of transformations.

Vertex

The vertex of a parabola is the point at which the function reaches its maximum or minimum value. It is the turning point of the graph and can be found using the formula x = -b/(2a) in the standard form of a quadratic function. Once the x-coordinate is determined, substituting it back into the function gives the corresponding y-coordinate.

Transcript

00:01 I am going to graph the quadratic function f of x equals 3x squared.

00:09 In addition to that, i am also going to describe some key features within that graph, such as the vertex, axis of symmetry, and the domain and range.

00:20 So the first thing i want to do is determine my vertex because that's the starting point of my parabola.

00:26 And based on my equation, i can tell that this graph has not been shifted left or right.

00:31 And it has not been shifted up or down.

00:33 And what that means is my vertex is going to be at zero, zero.

00:38 So i can place that on my graph.

00:40 And when i create a table for a quadratic function, i always put my vertex in the middle because it's a symmetric function, meaning i'm going to have the same things on the left and on the right.

00:52 And so i can build my table going up and down.

00:56 Okay.

00:57 Now, if you recall, the axis of symmetry is the x value of my vertex.

01:03 So in this case, my axis of symmetry is going to be x equals zero.

01:07 So there's, that's where that invisible line that cuts your parabola and half would go.

01:14 As i go to graph my parabola, i take a look at the three, which is my a value out in front.

01:20 And that's going to tell me that it's positive, so my graph should open up, and then it's going to be a little bit skinnier than my parent function.

01:27 When you're determining values for the function, we can, we can, try different values.

01:34 So i'm going to start with 1.

01:36 So when x is 1, so f of 1, equals 3 times 1 squared, well, we know 1 squared is 1.

01:45 So in this case, f of 1, 3 times 1, is 3.

01:50 Again, i mentioned that it's symmetric, which means if 1 is 3, then negative 1 is also 3...

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