What is a turning point on a quadratic graph?

Mathematics

1 answer:

Cerrena [4.2K]2 years ago

3 0

Answer:

A turning point is the highest or lowest point on a quadratic graph.

Step-by-step explanation:

A quadratic graph looks something like the graph below.

The equation of a quadratic graph would normally look like

+/- ax^2 + bx + c

An example might be -16x^2 + 5x + 4

Note the negative symbol in front of the 16. The negative means that the graph will be facing downwards, or that the turning point is the highest point. A positive graph will mean that the graph is facing upwards, or that the turning point is the lowest point.

Essentially, it is the location where a graph has its lowest or highest point and where the y-values (can include x-values in horizontal quadratics) "turn" to the direction they originated.

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The change in water level of a lake is modeled by a polynomial function, W(x). Describe how to find the x-intercepts of W(x) and

Agata [3.3K]

First. Finding the x-intercepts of $W(x)$

Let $W(x)$ be the change in water level. So to find the x-intercepts of this function we can use The Rational Zero Test that states:

To find the zeros of the polynomial:

$f(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{2}x^{2}+a_{1}x+a_{0}$

We use the Trial-and-Error Method which states that a factor of the constant term:

$a_{0}$

can be a zero of a polynomial (the x-intercepts).

So let's use an example: Suppose you have the following polynomial:

$W(x)=x^{4}-x^{3}-7x^{2}+x+6$

where the constant term is $a_{0}=6$ . The possible zeros are the factors of this term, that is:

$1, -1, 2, -2, 3, -3, 6 \ and \ -6$ .

Thus:

 $W(1)=0 \\ W(-1)=0 \\ W(2)=-12 \\ W(-2)=0 \\ W(3)=0 \\ W(-3)=48 \\ W(6)=840 \\ W(-6)=1260$ 

From the foregoing, we can affirm that $1, -1, -2 \ and \ 3$ are zeros of the polynomial.

Second. Construction a rough graph of $W(x)$

Given that this is a polynomial, then the function is continuous. To graph it we set the roots on the coordinate system. We take the interval:

$[-2,-1]$

and compute $W(c)$ where $c$ is a real number between -2 and -1. If $W(c)>0$ , the curve start rising, if not, the curve start falling. For instance:

$If \ c=-\frac{3}{2} \\ \\ then \ w(-\frac{3}{2})=-2.81$

Therefore the curve start falling and it goes up and down until $x=3$ and from this point it rises without a bound as shown in the figure below