Equations being described about Quadratic Function

[Joystar77]

These 3 equations all describe the same quadratic function. What are the coordinates of the following points on the graph of the function? From which equation is each point most easily determined?

y = (x - 5) (x + 1)

y = x ^ 2 - 4x - 5

y = (x - 2) ^ 2 - 9

X- intercept, what are the points, the equation, and explanation why that equation is the one from which the x-intercepts are most easily determined?

Is this the correct way to do this problem?

The simple way to graph y = (x - 5) (x + 1) is to generate at least 4 points, put those on graph paper and draw a straight line through them. Here's how I generate the required points: Use the equation y = (x - 5) (x + 1) and choose an integer for x, say x = 2, and substitute this into the equation to find the corresponding value of y.

y = (x - 5) ( x + 1)

y = (2 - 5) ( 2 + 1)

y = (-3) ( 3)

y = -9

So, my first point has coordinates of ( 2, -9). Now I would repeat this operation with a different value of x, say x = 4.

y = (x - 5) (x + 1)

y = (4 - 5) (4 + 1)

y = (-1) (5)

y = -5

So, my second point has coordinates of (4, -5). Now mark these two locations on graph paper. Starting at the origin of my graph (where the x-axis crosses the y-axis), go to the right 2 squares (x = 2) then down 9 squares (y = -9) and mark my first point. For the second point, again, I start at the origin and go right 4 squares (x = 4) and then down 5 squares (y = -5) and mark my second point. Using a straight-edge, draw a line joining these two points. I have now graphed the equation y = (x - 5) (x + 1). Compare the graph with the graph of y = (x - 5) (x + 1).

So far, is this correct? If not, am I thinking of something else? Can someone correct me on how to do this problem?

 

[stapel]

These 3 equations all describe the same quadratic function. What are the coordinates of the following points on the graph of the function? From which equation is each point most easily determined?

y = (x - 5) (x + 1)

y = x ^ 2 - 4x - 5

y = (x - 2) ^ 2 - 9

X- intercept....

I'm not sure why you're graphing...? You've been asked for the coordinates of the x-intercept(s), if any. What is the value of y at the x-intercept(s)? When y is that value, what is x? And (to answer the question in the exercise) with which equation is it simplest to solve for the value(s) of x? Why?

 

[Mrspi]

It appears that you've been studying various forms for the equation of a parabola, and you are being asked to recognize WHICH of those forms is the most convenient to use if you are dealing with a specific type of problem.

Here are the three most commonly used forms (they should be in your textbook, FOR SURE!):

Vertex Form

y = a(x - h)² + k

vertex is at (h, k)

Example: What is the vertex of the parabola whose equation is

y = 4(x - 3)² + 7

Compare to the general vertex form. Note that in this example, h = 3 and k = 7. So, the vertex is at the point (3, 7).

Intercept Form


y = p(x - a)(x - b)

x-intercepts occur at (a, 0) and (b, 0)

Note that whenever x = a or x = b, the value of y will be 0, and thus the point (a, 0) and the point (b, 0) will lie on the x-axis.

Example:

find the x-intercepts of the parabola with the following equation:

y = 2(x - 5)(x + 3)

Recall that (x + 3) can be thought of as (x - (-3)). So if x = 5, or if x = -3, y = 0. The x-intercepts are at (5, 0) and (-3, 0)

Standard Form

y = ax² + bx + c

The most useful information easily obtained from this form of the equation is the y-intercept; when x = 0, we get y = a(0)² + b(0) + c, or y = c. So the point (0, c) is on the y-axis and is the y-intercept for the graph.

Example:

Find the x-intercept of the graph of the parabola with this equation:

y = 3x² + 7x - 2

Since "c" in this equation is -2, the y-intercept of the graph would be (0, -2).

I hope this helps you.