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

Vertex Form

Vertex form of a quadratic function expresses the function as a shifted version of the basic quadratic y = a(x-h)^2 + k. This form clearly shows the vertex as (h, k) and emphasizes the horizontal and vertical shifts from the origin, making it easier to graph and analyze the function.

Vertex

The vertex of a quadratic function is the highest or lowest point on its graph, depending on whether the parabola opens downward or upward, respectively. It serves as the point of symmetry for the parabola and represents either the maximum or minimum value of the function.

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations of the form ax^2 + bx + c = 0. It provides the x-intercepts of the quadratic function by calculating the roots as x = (-b ± ?(b² - 4ac)) / (2a), which is particularly useful when the quadratic does not factor nicely.

Intercepts

Intercepts are the points where the graph of a function crosses the axes. In quadratic functions, the y-intercept is found by evaluating the function at x = 0, while the x-intercepts are the solutions to the quadratic equation, which can be determined using techniques such as factoring, completing the square, or the quadratic formula.

End Behavior

End behavior describes how the values of a quadratic function behave as the input values approach positive or negative infinity. For quadratic functions, the sign of the leading coefficient determines whether the parabola opens upward (positive coefficient) or downward (negative coefficient), which directly influences its long-term trend.

Graphing Quadratic Functions

Graphing quadratic functions involves plotting a parabolic curve using various key features such as vertex, intercepts, axis of symmetry, and end behavior. This process helps visualize the function's behavior, identify its maximum or minimum point, and understand its overall shape and orientation.

Completing the Square

Completing the square is an algebraic technique used to rewrite a quadratic function in vertex form. This method involves manipulating the quadratic into a perfect square trinomial plus a constant, which reveals the vertex coordinates directly and facilitates understanding of the function's translations and adjustments from the standard form.

Transcript

00:01 Hello.

00:02 So today we'll be working on solving quadratic equations and more specifically graphing this particular equation, y equals negative x squared plus 8x minus 7.

00:14 So to get started, we're going to complete the square for this equation in order to transform it into a better form for graphing.

00:28 So to complete the square, we've got y.

00:35 Equals negative.

00:37 I like having that a be positive when i'm working with it.

00:40 So we're just going to take the negative out of this whole thing, which will make that positive, make it negative 8x plus 7.

00:49 And then we'll multiply the negative back in at the end.

00:55 So here's x squared minus 8x.

01:02 And then the completing the square part, when you divide 8 by 2, you get 4.

01:09 4 squared is 16.

01:11 So you add 16 in here.

01:20 And then out here, remember we have positive 7 because our negative is out front.

01:26 And then we subtract 16 to keep the equation even.

01:32 So the equation has not changed since we added and then subtracted the 16.

01:37 We just changed how it looks.

01:41 So now y equals this x squared minus 8x plus 16 is going to simplify down to x minus 4 square.

02:01 And then that negative gets multiplied back to it.

02:04 And then 7 minus 16 is negative 9, multiplied by negative is positive 9.

02:14 So here's our equation.

02:16 Now to find the vertex, we're going to use the vertex formula, sorry about that.

02:24 The vertex formula to find the x value where the vertex occurs, which is h.

02:31 So h is equal to negative b over 2a.

02:43 Well, b here is 8, and a is negative 1.

02:49 So negative 8 over negative 2...

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