For quadratic function, identify the vertex, axis of symmetry, and x- and y-intercepts. Then, gr

For quadratic function, identify the vertex, axis of...

Key Concepts

Graphing Techniques

Graphing a quadratic function involves plotting the vertex, drawing the axis of symmetry, and marking key points such as the intercepts. Understanding the transformation of the basic quadratic graph, including shifts, reflections, and scaling, allows for a systematic approach to accurately drawing the parabola and analyzing its behavior.

Intercepts

Intercepts are the points where the graph of the quadratic function crosses the coordinate axes. The x-intercepts, found by solving the equation y = 0, indicate the roots or solutions of the quadratic function, while the y-intercept, obtained by setting x = 0, reveals the starting position of the graph on the y-axis.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror image halves. This line can be represented by the equation x = h when the quadratic function is in vertex form, and it is essential for understanding the symmetry properties of the graph.

Vertex Form

The vertex form of a quadratic function expresses the function as y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it easier to immediately identify the vertex and facilitates the graphing process by highlighting the transformation from the basic quadratic function.

Quadratic Function

A quadratic function is a second-degree polynomial function characterized by its general form ax² + bx + c, and its graph is a parabola. The quadratic function has properties such as symmetry, a vertex as its maximum or minimum point, and distinct intercepts where it crosses the axes, making it an important object of study in algebra.

Vertex

The vertex is the turning point of a parabola, representing either the maximum point (if the coefficient of the squared term is negative) or the minimum point (if it is positive). It is crucial in graphing the quadratic function and understanding its behavior since the vertex determines the direction of opening and is used to construct the axis of symmetry.

Transcript

0:00 Right.

00:01 So we are given the function here, y, is equal to 2 times x plus 1 squared minus 2.

00:16 Okay, and the first we want to find the vertex.

00:20 The vertex, well, we have a quadratic in vertex form, a times x minus h squared plus k.

00:28 So the vertex is just the point h comma k.

00:31 What is h? well, we have x plus one, which is x minus a negative one.

00:37 So h is a negative one.

00:40 And we're always plus k.

00:41 We have minus two.

00:43 That's plus a negative two.

00:45 So k is a negative two.

00:47 So our h is negative one and our k is negative two.

00:52 It's always a times x minus h squared plus k.

00:57 This is our vertex.

00:58 Our axis of symmetry is just the line that goes right through the vertex.

01:05 So our axis of symmetry is the vertical line, which is always x equals and just the value of our x value in our vertex, which is negative one.

01:15 So our axis of symmetry is just the vertical line, x equals negative one.

01:21 All right.

01:22 It's a vertex of axis of symmetry, and then our x intercepts, our x intercepts, intercepts is where we cross the x -axis.

01:32 Now we cross the x -axis, the value of y there is equal to zero.

01:38 Right? we have x comma zero.

01:42 So we just set our function.

01:46 We can set our function or set y equal to zero and find the value of x that makes the function equal to zero.

01:53 That'll be your x intercepts.

01:54 So to find your x intercepts, you set the function equal to zero and then solve for x so we have zero equals two times x plus one squared minus two so we got to add two to both sides here to get two equals two times x plus one squared by adding to both sides then i have a coefficient of two that can divide through by so divide through by two two over two comes one it's cancel is here, and then 2 over 2 is 1.

02:31 So i get 1 equals x plus 1 squared, dividing both sides by 2.

02:41 And then taking the square root of both sides, well, the square root of 1 is, well, plus or minus 1.

02:49 Okay? because, well, if i think the square of 1, you say square of 1 is 1, okay? and the square root of x plus 1 squared is just x plus 1.

02:59 The square root of something squared is just, if they can't, the square root can't start the square, i'm just left with x plus one, but also negative one, right? it's always at this step, either plus or the mind, because negative one squared is also one.

03:16 So, get the square root of one, you get plus or minus one.

03:22 So i get minus one is also equal to x plus one...

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