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.


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


c) What are the coordinates of the vertex?

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

we get


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)


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



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.


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, ∞)


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


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)


8 graph 2x


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


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