2x 8 graph

2x 8 graph DEFAULT

5. Draw the graph of y = f(x) = x2 + 2x - 8.

 

  • a) What is the y-intercept?
  • b) What are the roots?
  • c) What are the coordinates of the vertex?
  • d) Write the equation in standard or vertex form.
  • e) Solve for x

Before we draw the graph it would help to answer questions a), b), and c).

a) What is the y-intercept?

The y-intercept is the point on the graph whose x-coordinate is 0. So let x = 0 in the equation, and we get

This illustrates that the y-intercept is always the constant term in a polynomial.

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b) What are the roots?

The roots are another name for the x-intercepts. They are the points on the graph whose y-coordinates are 0. So let   y = 0   in the equation and we get

This gives us a quadratic equation in   x. &mbsp; We are fortunate enough to already have a 0 on one side so we are ready to factor.

Set the factors = 0.

Solve for x

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c) What are the coordinates of the vertex?

If we use the formula for the x-coordinate of the vertex,

we get

or

Now that we have the x-coordinate of the vertex, we run this number through the function to find the y-coordinate

 

So the vertex is at (-1, -9)

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It is a good idea to get all of this information before plotting points to draw the graph. When we plot points, since we now know that the x coordinate of the vertex is -1, we will want -1 to be in the middle of the x's that we plot. We will also want the roots,   x = -4   and   2   to be in the interval containing the x's that we plot, and at least one point on the other sides of the roots from the vertex.

When we plot these points, we get

 

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d) Write the equation in vertex form.

The equation is given to be

 

Half of the linear coefficient is 1, and the square of 1 is 1, so we add and subtract 1

This simplifies to

 

and we see the coordinates of the vertex in the equation.

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e) Solve for   x

In the process of drawing the graph we found that the roots were at -4 and 2. When we divide the real number line into the intervals between the roots, and check out an x in each interval.

Since equality is permitted, the endpoints of the intervals, the roots, being the x's which will give answers which are equal to 0 when substituted into the formula, will be solutions. So we draw square brackets around the endpoints.

The set of solutions is the set of   x's   which satisfy

x ≤ -4   or   x ≥ 2

In interval notation, this comes out to be

(-∞, -4] ∪ [2, ∞)

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Return to test

 

 

Sours: http://web.sonoma.edu/users/w/wilsonst/courses/math_50/SolnsS98/MT2/p3/MT2S98Soln3.html

Properties of a straight line

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     y-(2*x-8)=0 

Step  1  :

Equation of a Straight Line

 1.1     Solve   y-2x+8  = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  y-2x+8  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is -8/1 so this line "cuts" the y axis at y=-8.00000

  y-intercept = -8/1 = -8.00000

Calculate the X-Intercept :

When y = 0 the value of x is 4/1 Our line therefore "cuts" the x axis at x= 4.00000

  x-intercept = 8/2 = 4

Calculate the Slope :

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is -8.000 and for x=2.000, the value of y is -4.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of -4.000 - (-8.000) = 4.000 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

  Slope = 4.000/2.000 = 2.000

Geometric figure: Straight Line

  1.   Slope = 4.000/2.000 = 2.000
  2.   x-intercept = 8/2 = 4
  3.   y-intercept = -8/1 = -8.00000
Sours: https://www.tiger-algebra.com/drill/y=2x-8/
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Sketching a nice graph of $f(x) = x^2 + 2x – 8$ [closed]

So the intercepts first.

When $x=0$ you have $f(x)=-8$ so the point $(0,-8)$ is on the graph.

When $f(x)=0$ you have $x^2+2x-8=0$ and you can see from the graph you have, or from solving the quadratic, or by factoring as $(x+4)(x-2)=0$ that this happens at the points $(-4,0)$ and $(2,0)$.

The vertex is the maximum or minimum point - now this can be done using calculus, but completing the square is as easy for a quadratic. I'll leave you to see whether you can identify the relevant point, and determine whether it is a maximum or minimum, if the function is rewritten $f(x)=(x+1)^2-9$.

That gives you four key points, and together with the knowledge that the graph is a parabola, you should be able to construct a decent sketch.

answered Jun 22 '15 at 15:45

Mark BennetMark Bennet

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$\endgroup$Sours: https://math.stackexchange.com/questions/1335047/sketching-a-nice-graph-of-fx-x2-2x-8
Domain and Range of 1/(x^2-2x-8)

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8 graph 2x

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Draw the graph of the polynomial `f(x)=x^2-2x-8`

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