Graphs of Quadratic Functions The graph of a quadratic function f is given. (a) Find the coordin

Graphs of Quadratic Functions The graph of a quadratic...

Key Concepts

Domain and Range

The domain of a quadratic function is all real numbers, as there are no restrictions on the values x can take. The range, however, depends on the vertex and the direction the parabola opens. If the parabola opens upward, the range starts at the minimum value (the y-coordinate of the vertex) and extends to positive infinity; if it opens downward, the range extends from negative infinity up to the maximum value at the vertex. Understanding the domain and range is essential for comprehending the complete set of inputs and outputs of the function.

Extremum (Maximum/Minimum) of a Quadratic

A quadratic function always has either a maximum or a minimum value, occurring at the vertex. If the coefficient 'a' is negative, the parabola opens downward and the vertex represents the maximum point. Conversely, if 'a' is positive, the parabola opens upward and the vertex is the minimum point. This extremum is a key characteristic in analyzing quadratic functions, as it often represents the optimal value in real-world applications.

Vertex of a Quadratic

The vertex of a quadratic function is the point where the parabola attains its maximum or minimum value. It serves as the turning point of the graph. The coordinates of the vertex can be found using the formula (-b/2a, f(-b/2a)), or by rewriting the quadratic in vertex form via completing the square. The vertex is a crucial point as it provides important information about the function's extremum and overall behavior.

Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form f(x) = ax^2 + bx + c. These functions graph as parabolas, and their overall shape and direction (opening upward or downward) are determined by the coefficient 'a'. Quadratic functions appear in many applications across mathematics, physics, and engineering, often modeling scenarios involving acceleration or projectile motion.

Intercepts of a Function

Intercepts are the points where the graph of the function intersects the coordinate axes. In the context of quadratic functions, the y-intercept is found by evaluating the function at x = 0, and the x-intercepts (or roots) are the values of x for which f(x) = 0. Finding these intercepts assists in graphing the function and understanding its interaction with the axes.

Transcript

00:01 Okay, so this problem asks us to find the coordinates of the x and find the coordinates of the vertex.

00:08 And the way we do that is by putting you a standard form.

00:11 And the way we do put it in the standard form is by completing the square.

00:17 So the first thing i'll do is take out the negative one -half common from the first two terms.

00:22 So i get negative one -half times x squared plus 4x plus plus, 6 and i want to complete the square.

00:34 So how i do that is i take the b term divided by 2 and square it for my c term.

00:39 So i get negative 1 half x squared plus 4x plus 4.

00:50 And since i'm adding 4 times negative 1 half, negative 2 to the equation, i must add 2 to the equation to make it equal.

00:58 So i have plus 6 plus 2.

01:01 So now both these statements are equal.

01:04 And i'm just going to simplify this further to put it in standard form, x plus 2 squared plus 8.

01:17 All right, now it's in standard form.

01:21 I can find that is the answer.

01:23 So now it's in standard form, right? part a asks us to find the coordinates of the vertex, right? and that is just h comma k, which is negative 2, comma, 8.

01:37 Part b s is to sorry, part a asks us to also find the x and y intercepts.

01:46 So the y intercept is just when x is equal zero.

01:49 So we plug in zero for x we get the y intercept as six...

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