Quiz Quiz on graphing linear and quadratic graphs

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Question: Without drawing a graph how can I determine (a) how many x-intercepts the parabola has (b) whether its vertex is above, below or on the x- axis?

y = -x^2 + 2x-1
Answers:

You can put this solution on YOUR website!
x intercepts is just another way of saying "roots". Find roots of this quadratic expression with y set to 0.
since coefficient in front of x is negative, the parabola is pointing downwards.
where its vertex is, depends on the roots.
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case -1x%5E2%2B2x%2B-1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A-1%2A-1=0.

Discriminant d=0 is zero! That means that there is only one solution: x+=+%28-%282%29%29%2F2%5C-1.
Expression can be factored: -1x%5E2%2B2x%2B-1+=+-1%28x-1%29%2A%28x-1%29

Again, the answer is: 1, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B2%2Ax%2B-1+%29


since both roots are same, its vertex is right on the x axis.
Question: here is another question i have. the directions say, "write an equation in slope-intercept form that satisfies each condition." and here is a problem. "slope=-0.5, passes through (2,-3)". i know that the answer will have something like this= y=-0.5x because the slope is given so thats next to the "x". but whats with the rest? thank you so much whoever gets to answer.
Answers:

You can put this solution on YOUR website!
you are correct so far, but you have forgotten the standard form of a straight line... y=mx+c

so you have y=-0.5x+c --> need to find c. To do this, you need to know x and y at a point..you do, namely (2,-3)...so put those in the equation, to find c.

Hope that helps :-)

jon.
Question: Determine the slope and y-intercept of the graph of 7x-3y=2.
Answers:

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Since the slope-intercept form of the equation of a line gives you the slope and intercept directly, your job is to rearrange the equation into the form
y+=+mx+%2B+b where m is the slope and b is the y-intercept.
7x-3y=2
-3y=2-7x [subtract 7x from both sides]
y=%282%2F-3%29+-+%287x%2F-3%29 [divide both sides by -3]
y=%287%2F3%29x+-+2%2F3 [rearrange to match slope-intercept form]
slope = 7/3
y-intercept = -2/3
Question: Can someone help with question:
Graph the parabola:
y=(x-3)^2-3
Answers:

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right then, y=%28x-3%29%5E2-3 is a u-shaped curve.

When x=0, y=%28-3%29%5E2+-+3 --> y=9-3 --> y=6

When y=0, %28x-3%29%5E2+-+3+=+0
%28x-3%29%5E2+=+3
x-3 = +- sqrt%283%29
so either x = 3%2Bsqrt%283%29
or x = 3-sqrt%283%29

this gives you enough info to sketch the curve.

If you want to PLOT it properly, then pick various x values and find the y-values then plot them.

jon.
Question: May someone help me understand this problem. I have looked in my testbook and I can not find an example that can help me solve this problem.
The question is:
Solve the following sysstem of equations by graphing:
y = x^2 - 4 and y = x - 2.
Thanks,
Angie
Answers:

You can put this solution on YOUR website!
graph%28+500%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2-4%2C+x-2+%29+
the graphs intersect in points where x and y values are the same for both functions. So we can write
x^2-4=x-2
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • This is an equation! Solutions: x=2,x=-1.
  • Graphical form: Equation x%5E2-4=x-2 was fully solved.
  • Text form: x^2-4=x-2 simplifies to 0=0
  • Cartoon (animation) form: simplify_cartoon%28+x%5E2-4=x-2+%29
    For tutors: simplify_cartoon( x^2-4=x-2 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at x%5E2-4=highlight_red%28+x-2+%29.
Moved these terms to the left highlight_green%28+-x+%29,highlight_green%28+2+%29
It becomes x%5E2-4-highlight_green%28+x+%29%2Bhighlight_green%28+2+%29=0.
Look at x%5E2-highlight_red%28+4+%29-x%2Bhighlight_red%28+2+%29=0.
Added fractions or integers together
It becomes x%5E2%2Bhighlight_green%28+-2+%29-x=0.
Look at x%5E2%2Bhighlight_red%28+-2+%29-x=0.
Moved -2 to the right of expression
It becomes x%5E2-x%2Bhighlight_green%28+-2+%29=0.
Look at x%5E2-x%2Bhighlight_red%28+-2+%29=0.
Removed extra sign in front of -2
It becomes x%5E2-x-highlight_green%28+2+%29=0.
Look at highlight_red%28+x%5E2-x-2+%29=0.
Equation highlight_red%28+x%5E2-x-2=0+%29 is a quadratic equation: x^2-x-2 =0, and has solutions 2,-1
It becomes highlight_green%28+0+%29=0.
Result: 0=0
This is an equation! Solutions: x=2,x=-1.

Universal Simplifier and Solver


Done!
Question: Dear Sir/Madam,
I am confronted with the following problem:
"Where does the graph of y = 2 + 2^x cross the y-axis?"
Is there a way of solving this without graphing the function?
Thanks in advance.
Regards,
-Mike
Answers:

You can put this solution on YOUR website!
"Cross the Y axis" means the point where x = 0. So, use 0 for x in 2+2^x and you can find out the Y point. 2+2^0 is 2+1=3, so the point is (0, 3).
+graph%28+400%2C+300%2C+-5%2C+3%2C+-2%2C+10%2C+2%2B2%5Ex%29+

You can put this solution on YOUR website!
y=2+2^(x)
To find the y-intercept, substitute in 0 for x and solve for y.
y=2+2^(0)
Any number raised to the power of 0 is 1.
y=2+1
Solve the equation.
y=3
These are the x and y intercepts of the equation y=2+2^(x).
y=3
It crosses the y intercept at 0,3
To graph this equation you must also solve for x.
The x intercept is 1,0
Question: Please assist me in comprehending how to solve this problem.
Find the solution of the system y = x^2 + 5 and y = 2x + 4
Answers:

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y = x^2 + 5
y = 2x + 4
---
Therefore,
x^2 + 5 = 2x + 4
x^2 - 2x + 5 - 4 = 0
x^2 - 2x + 1 = 0


Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B1+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-2%29%5E2-4%2A1%2A1=0.

Discriminant d=0 is zero! That means that there is only one solution: x+=+%28-%28-2%29%29%2F2%5C1.
Expression can be factored: 1x%5E2%2B-2x%2B1+=+1%28x-1%29%2A%28x-1%29

Again, the answer is: 1, 1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B1+%29



Therefore,x=1
then y=2x+4=2+4=6


x=1,y=6

Hope this helps,
good luck
Question: draw the graph of 2x+5y=8.
I don't know how to find x and y . Could you help?
Answers:

You can put this solution on YOUR website!
In these kinds of problems, you assign values of your own choosing to x (the independent variable) and then you compute the corresponding value of y (the dependent variable). You can then make what is called an input-output table showing these values. It makes things a little easier if you first solve the equation for y by putting it into the slope-intercept form, y = mx + b.
We'll do this for this equation, 2x + 5y = 8
2x + 5y = 8 Subtract 2x from both sides.
5y = -2x + 8 Divide both sides by 5
y = (-2/5)x + 8/5
Now set up the input-output table. The x-value that you assign corresponds to the input and the resulting y-value is the output.
x, y
--------
0, 8/5
1, 6/5
2, 4/5
-1, 10/5
-2, 12/5
Of course, you need only two points to define a straight line but you can calculate any number of points you wish.
Now, on a piece of graph paper on which you have drawn the x-axis (horizontal) and the y-axis (vertical), you can graph some or all of the points in the table and then join them together with a straight line. It should look like the graph shown below:


graph%28300%2C200%2C-10%2C10%2C-10%2C10%2C%28-2%2F5%29x%2B8%2F5%29
Question: what does the inequality 2x-4y>-6 look like on a graph?
Answers:

You can put this solution on YOUR website!
first find what 2x-4y=-6 looks like, since this will be a straight line you can plot/sketch. Then 2x-4y>-6 will be either the region above the line or the region below the line.

So, to stech the line, we need it in the form y=mx+c, so:

2x-4y=-6
-4y=-2x-6 or...
4y = 2x+6
y = (2/4)x + 6/4
--> y = (1/2)x + 3/2

so, we know the gradient is +1/2
and we know the line crosses the y-axis at y=3/2.

So, sketch that.

Now for your question...where is 2x-4y>-6? well, following through the same process:

2x-4y > -6
-4y > -2x-6 or...
4y < 2x+6 -- since we multiplied all terms by -1 (to swap all signs round) we swap the > into a <
y < (2/4)x + 6/4
--> y < (1/2)x + 3/2

so, where is y less than the straight line? Answer: below the line.

So we want the region below the line.

Jon.
Question: what does 2x+y=-5 look like on a graph?
Answers:

You can put this solution on YOUR website!
It is a line with a slope of -2, which intersects the y axis at the point (0,-5)
I simply put the equation in the format y=mx+b, where the slope is m and the y-intercept is b
y=mx+b
2x+y=-5
subtract 2x from both sides:
y=-2x-5
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