Write the quadratic function in standard form and sketch its graph. Identify the vertex, axis of

Write the quadratic function in standard form and sketch its...

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function of degree two, generally expressed as f(x) = ax² + bx + c, where a, b, and c are constants and a is not zero. This form provides a clear framework for analyzing the shape and key characteristics of the function, such as its vertex and intercepts.

Standard Form

The standard form of a quadratic function is f(x) = ax² + bx + c. This form makes it easy to identify the coefficients and supports further analysis, including methods like the quadratic formula, which are used for finding the x-intercepts of the function.

Vertex

The vertex of a quadratic function is the point where the function attains its maximum or minimum value. It is a crucial feature because it represents the turning point of the parabola. The x-coordinate of the vertex can be found using -b/(2a), and substituting this back into the function gives the y-coordinate.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex and divides the parabola into two mirror-image halves. This line has the equation x = -b/(2a), indicating that the parabola is symmetric about this line, which aids in graph sketching by providing a symmetry guide.

X-intercepts

X-intercepts are the points at which the quadratic function crosses the x-axis, meaning where the function equals zero. They are found by solving the quadratic equation ax² + bx + c = 0, using methods such as factoring, completing the square, or the quadratic formula.

Graphing Quadratics

Graphing quadratic functions involves plotting the vertex, drawing the axis of symmetry, and identifying the x-intercepts by solving f(x) = 0. After these key features are located, additional points can be computed if necessary. The overall graph is a parabola that either opens upward or downward, depending on the sign of the coefficient a.

Transcript

00:01 Okay, this question asks us to take a particular function, in this case, x squared minus 6x, and write it in standard form.

00:08 Now, if you recall, standard form is a function that's quadratic of the form x minus x0 squared plus k, where this x knot and k are just numbers.

00:23 So to get this function in that form, what we're going to do is complete the square.

00:29 So i'm going to add 9 to both sides of this equation.

00:34 What we end up with is f x plus 9 equals x squared minus 6x plus 9.

00:41 Now we know that this here is a perfect career, and it is x minus 3 quantity squared.

00:49 Now all we have to do is move this 9 over.

00:52 So rewritten, what we have is f of x is equal to now x minus 3, quantity squared.

00:59 V squared minus nine.

01:03 Now this question also asks us to sketch the graph and find its intercepts and its vertex.

01:12 So let's sketch the graph.

01:15 Let's take these to be units of three and these to be units of one, which will ultimately give our graph a kind of squished appearance, but it will largely look the same.

01:29 So let's start with x equals 0.

01:31 Well 0 minus 3 is negative 3.

01:33 Negative 3 squared is 9.

01:35 So 9 minus 9 will give us that 0 back.

01:39 So let's take 1.

01:41 1 minus 3 is negative 2.

01:42 Negative 2 squared is 4.

01:45 4 minus 9 is negative 5.

01:49 5...