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

Name: Problem 42, Chapter 11: Beginning and Intermediate Algebra
Uploaded: 2022-02-16T19:46:01-08:00
Duration: 3 min 46 s
Channel: Jessica Kluck
Description: Graph each parabola. Give the vertex, axis of symmetry, domain, and range. f(x)=-2(x-2)^2-3

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

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree 2 that graph as parabolas. They are foundational in algebra and calculus, and their standard form is ax^2 + bx + c. Understanding quadratic functions provides insight into concepts such as turning points, symmetry, and the relationship between algebraic expressions and their geometric representations.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is written as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify translations and dilations of the parent function f(x) = x^2, helping to quickly determine key graphing attributes such as the vertex and the direction in which the parabola opens.

Vertex

The vertex is the highest or lowest point on a parabola, depending on whether it opens downward or upward. In the vertex form of a quadratic function, the vertex is directly given by the point (h, k). It is a crucial concept in understanding the behavior of quadratic functions, particularly in optimization problems.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in vertex form, the axis of symmetry is the line x = h, where h is the x-coordinate of the vertex. This concept is essential for analyzing the properties of parabolic graphs and solving related equations.

Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For quadratic functions, the domain is typically all real numbers, reflecting the fact that there are no restrictions on x in a quadratic equation.

Range

The range of a function is the set of all output values (y-values) produced by the function. In the case of quadratic functions, the range is determined by the vertex and the direction in which the parabola opens. This concept is important for understanding the limitations and possible values a quadratic function can output over its domain.

Transcript

00:01 I want to graph and identify some key features of the quadratic function f of x equals negative 2 times x minus 2 squared minus 3.

00:09 To begin, i'm going to notice that it is in vertex form.

00:13 And so i'm going to locate my vertex.

00:16 My x minus h means my h is going to be positive 2, meaning i'm going to the right 2.

00:23 And my k is negative 3, so i'll be going down 3 units.

00:28 I do know that my axis of symmetry is the x value of my vertex, so x equals 2.

00:34 And i'm going to go ahead and throw 2 negative 3 into the middle of my table because parabolas are symmetric.

00:40 And that means i'm going to have the same on the left as i am on the right.

00:45 And i'll plot the point 2 -93 on my graph, dotting in my axis of symmetry.

00:54 Before i finish my table, i like to take a look at my a value.

01:00 And my a value is going to tell me a couple of things.

01:02 It's going to tell me that a is less than zero, meaning my parabola is going to open down.

01:12 And i also notice that the absolute value of a is greater than one, meaning it's going to be narrower than my parent function.

01:20 So quite a few of my values, it looks like, are going to probably fall off my coordinate plane i have set up here.

01:26 So i'm going to go ahead and surround my x value of two, on both directions.

01:35 And i'm going to evaluate for those x values so i can find my f of x.

01:40 So we'll start with f of 3.

01:44 And i'm going to take and plug the 3 in for x.

01:47 So i've got negative 2 times 3 minus 2 squared minus 3.

01:53 And negative 2 times 3 minus 2 is 1.

01:57 1 squared is 1 minus 3...

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