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

End Behavior

End behavior describes how the values of a quadratic function behave as x approaches positive or negative infinity. For a quadratic function, this behavior is determined by the sign of the leading coefficient: if a > 0, the arms of the parabola rise upward, and if a < 0, they fall downward. Understanding end behavior is essential for accurately sketching the graph of the function.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), provides a method for solving quadratic equations and finding the x-intercepts of the graph. It is particularly useful when the quadratic does not factor easily, offering a direct route to the solutions of ax² + bx + c = 0.

Intercepts

Intercepts are the points where a graph crosses the coordinate axes. The y-intercept is found by evaluating the function at x = 0, while the x-intercepts (or zeros) are determined by solving the equation f(x) = 0. These points are crucial for sketching the overall shape and position of the graph.

Quadratic Functions

A quadratic function is a second-degree polynomial, typically written in the form f(x) = ax² + bx + c. Its graph is a parabola, which opens upward if a > 0 and downward if a < 0. Understanding the general behavior of quadratic functions is fundamental, as it sets the stage for analyzing their specific features, such as intercepts, vertex, and symmetry.

Completing the Square

Completing the square is an algebraic technique used to convert the standard form of a quadratic function into vertex form. This process involves creating a perfect square trinomial from the quadratic and linear terms, allowing the function to be rewritten as a shifted square term plus or minus a constant. It is an essential tool for analyzing and graphing quadratic functions.

Vertex Form

Expressing a quadratic function in vertex form, f(x) = a(x - h)² + k, makes it easy to identify the vertex of the parabola at (h, k). This form arises from completing the square and is particularly useful for graphing because it clearly shows the horizontal and vertical shifts from the parent function y = x².

Graph Transformations

Graph transformations refer to the shifts, reflections, and dilations of the basic parabola y = x². In the context of quadratic functions, these transformations include horizontal and vertical shifts that position the vertex at a new location, reflections which occur when the leading coefficient is negative, and stretching or compressing depending on the magnitude of that coefficient.

Transcript

00:02 Hello, so today we're going to be going over how to graph a quadratic equation, specifically this equation, y equals negative 3x squared plus 12x minus 7.

00:15 So to start with, we're going to transform this function to complete the square.

00:23 This will make it easier to graph it.

00:27 So we're going to start, and i always find, it easier to take the coefficient in front of the x squared value out when completing the square.

00:42 So this becomes negative 3 times x squared minus 4x.

00:52 And then we complete the square in the parentheses here.

00:55 So negative 4 divided by 2 is negative 2.

00:58 Negative 2 squared is positive 4.

01:01 So we add four here.

01:07 And then you have your minus 7 remains separate.

01:12 And then this plus 4 we have in here.

01:17 But what we actually did, we didn't add 4.

01:19 We subtracted 12 because you still have to multiply by this negative 3 value.

01:29 So because of that, we have to add 12 over here in order to keep the equation equal to what it was before.

01:37 Which is important when completing the square, you don't want to change the equation at all.

01:43 So now we're going to go y equals negative three, and we'll simplify this parentheses down to x minus 2 squared.

02:02 And then minus 7 plus 12 is just plus 5.

02:06 All right, so here's our simplified equation now that we've completed the square.

02:12 Now to find the vertex of this graph, the x value of the vertex at h, according to the vertex formula, is at negative b over 2a.

02:28 So that is equal to negative 12 over negative 6, which is equal to negative 12, over negative 6, which equal to 2 that's 12 and 6 comes from our first equation before we transformed it the 12 here is b negative 3 is a it's important to make sure that you keep the negative with the coefficients so negative 3 times 2 is negative 6 negative 12 divide you get 2 so your x value for the vertex is at 2 this you can then plug into here at this x value.

03:15 So you get negative 3 times 0 because 2 minus 0 plus 5.

03:20 So k or the y value of the vertex is equal to 5.

03:30 Now because we now know that our vertex is going to be in quadrant 1 at 2 .5.

03:39 And our a value is negative, so it's going like this, you're going to have two x intercepts...

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