Graph each function using end behavior, intercepts, and completing the square to write the funct

Graph each function using end behavior, intercepts, and...

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial typically expressed in the form ax^2 + bx + c and graphs as a parabola. It is important because its characteristics, such as the vertex, axis of symmetry, and intercepts, are fundamental in algebra and calculus for understanding the behavior of polynomial functions.

Vertex Form

The vertex form of a quadratic function is written as a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. Rewriting a quadratic into this form makes it easier to identify the vertex and clearly see the horizontal and vertical shifts applied to the basic parabola y = x^2.

Completing the Square

Completing the square is the algebraic technique used to convert a quadratic function from standard form into vertex form. This process involves creating a perfect square trinomial from part of the quadratic, which then allows the function to be rewritten in a form that readily shows its transformations and vertex.

Transformations

Graph transformations refer to the shifts, reflections, stretches, or compressions applied to the parent function to obtain the graph of the given function. For quadratic functions, determining the horizontal shift, vertical shift, and reflection (if the coefficient of x^2 is negative) provides a clear path to sketching the graph accurately.

Intercepts

Intercepts are the points where the graph crosses the axes. The y-intercept is found by evaluating the function at x = 0, and the x-intercepts are found by solving the equation f(x) = 0. Identifying these intercepts is crucial because they offer key reference points for accurately sketching the graph of the function.

Quadratic Formula

The quadratic formula, which is x = (-b ± ?(b^2 - 4ac)) / (2a), provides a method for finding the x-intercepts (roots) of any quadratic equation. It is especially useful when the quadratic does not factor neatly, ensuring that all real or complex roots can be determined.

Transcript

00:01 So here we have a function that we're going to complete this square and using that we'll graph it and find the x intercepts, y intercepts, and the important parts of the graph.

00:12 So we'll start by completing the square.

00:14 We're going to separate this function into two different parts.

00:20 One with the variables, x, and then also a separate part for the constant.

00:27 So let's go ahead, negative 2x squared plus 8x.

00:33 We're going to separate it and add a constant so that we can complete the square.

00:38 And then we'll have the plus 7 outside of that.

00:42 We want to factor out the negative 2 to make it easier to work with.

00:46 Let's go ahead, negative 2 to x squared, then minus 4x plus that constant, plus 7.

00:55 Now i figure out what this constant is.

00:57 We'll use a simple formula of b over 2 squared.

01:00 So we have b as that 4, negative 4x.

01:06 We'll go ahead, negative 4 over 2, 2 squared, 4.

01:14 So we're going to add that 4.

01:19 And we can't forget the negative 2 that we fact it out earlier.

01:23 So in total, on this left side of the separation, we have added a negative 8.

01:29 So we need to subtract a negative 8 from the other side.

01:32 Side.

01:33 We're going to keep that function the same and not change any values.

01:38 So now that we've done this, we can go ahead to the next step of completing the square and factor out.

01:46 And we can find that it was x plus, sorry, x minus 2 squared.

01:55 And then we can add those two together, minus a minus cancel each other out, plus 15.

02:06 Now we can use this.

02:07 Now we can use this.

02:08 This as a transformation of the original function to in order to graph it.

02:14 And we can actually find our vertex just by this information.

02:18 So we're going to know that this graph is going to be a quadratic...

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