Finding Quadratic Functions Find a function f whose graph is a parabola with the given vertex an

Finding Quadratic Functions Find a function f whose graph is...

Key Concepts

Vertex Form of a Quadratic Function

Vertex form is a way of expressing a quadratic function as f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. This form is especially useful when the vertex is known, as it directly identifies the parabola's turning point and simplifies the process of constructing the function.

Substitution to Determine Unknown Parameters

When a quadratic function in vertex form has an unknown coefficient 'a', and an additional point on the graph is provided, substituting the coordinates of that point into the function allows for solving for 'a'. This step is essential for specifying the exact multiplication factor and, consequently, the width and direction of the parabola.

Properties of Parabolic Graphs

Understanding the vertex, the value of 'a', and the resulting coefficients helps in predicting the shape and orientation of the parabola. The vertex provides a key point that anchors the graph, while the sign and magnitude of 'a' determine whether the parabola opens upward or downward and how 'steep' the curve is.

Transcript

0:00 Tricky question.

00:01 This question asks us to find a function f whose graph meets a certain vertex and a parabola.

00:10 So it meets and the vertex 2 minus 3 and passes through the point 3 .1.

00:26 Okay, so if it starts saying anything about vertex, i instantly think of completing the square or standard form.

00:33 So what i'm going to do is i'm going to start off with my equation in standard form and i'll use a's and x's to represent that.

00:40 So it's going to look something like a times by x plus b all squared and it's going to be minus or plus i said probably plus a c as well.

00:55 So that's that in that form.

01:00 And then very similar to how we'd get the vertex out of it.

01:04 We're going to put the into it.

01:06 So if you imagine trying to extract a vertex from the equation in standard form, this b here is going to be this two but the opposite side and this c would be the minus three.

01:15 So the equation is actually going to be this, using the vertex, it's going to be this.

01:23 R squared minus three.

01:27 Okay, so we've got that now and now we only have one more unknown.

01:30 So if we only have one unknown and we have a point, then we can sub in the value from this point to get our no.

01:36 So i'm going to put in sub in three.

01:42 I'll do this in green here.

01:43 So y equals a.

01:47 Oh sorry.

01:48 So the y here is actually going to equal one...