The graph of f(x)=3(x-2)^2-6 is a parabola that opens , with its vertex at ( , ) and f(2)= i

The graph of f(x)=3(x-2)^2-6 is a parabola that opens ,...

Key Concepts

Direction of Opening of a Parabola

The coefficient 'a' in the quadratic function determines the direction in which the parabola opens. If 'a' is positive, the parabola opens upward, which means that the vertex is the minimum point. If 'a' is negative, the parabola opens downward, and the vertex is the maximum point.

Vertex of a Parabola

The vertex of a parabola is the point where the function takes on its extreme value, either a minimum or a maximum, depending on the value of 'a'. In the vertex form, this point is directly given by (h, k), making it easy to locate on the graph.

Quadratic Function

A quadratic function is a polynomial of degree two that can be expressed in the form f(x) = ax² + bx + c. Its graph is a parabola, and the function's form can reveal important properties such as the vertex, direction of opening, and axis of symmetry.

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 is particularly useful because it makes it straightforward to identify the vertex and determine whether the function has a maximum or minimum value.

Transcript

00:01 So for this question, it's asking us to read through a sentence and to fill in the blanks.

00:04 And it's asking us about certain aspects of a parabola fx.

00:07 The graph of fx is a parabola that opens, and it's asking us here whether it opens upwards or downwards.

00:13 And the way we can figure that out is if we just expand out fx, we're looking for what comes before the x squared or the coefficient of x squared.

00:21 And if the coefficient of x squared is positive, that means it goes upwards.

00:24 And it's negative, that means it goes downwards.

00:26 You can just do this in your head here.

00:30 So when we expand out this bracket, we're going to get just a single x -squared term and that's going to get tithed by 3, and that's going to be 3x squared.

00:37 You can take this in as many steps as you like, and expand it out as slowly as you want.

00:43 But the coefficient here, you're going to get 3.

00:45 And that is a positive coefficient.

00:48 So that means the graph opens up upwards.

01:00 There you go.

01:02 Okay, so then it's asking for the vertex.

01:05 And the way you found out the vertex, if we can just read this off of the standard form version of fx.

01:11 So for our x ordinate, it's just going to be the positive, not the positive, the opposite sign version of what comes after the x inside of the bracket.

01:20 So that's minus two, but since we need the opposite sign, it's positive two for the x ordnate.

01:27 And if you have any questions about where that comes from, you can email me and i'll, my email at the end...

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