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3 years ago

Find the vertex of the graph of the function. f(x) = (x + 4)2 - 1

Mathematics

1 answer:

Elanso [62]3 years ago

5 0

Answer:

The vertex of the function is at point (-4,-1).

Step-by-step explanation:

Given function:

$f(x)=(x+4)^2-1$

Solution:

The vertex form of a function is given by:

$f(x)=a(x-h)^2+k$

where $(h,k)$ is the vertex of the function. At this point the function has the maximum or minimum value.

Writing the given function in the vertex form.

$f(x)=(x-(-4))^2+(-1)$

On comparing the above function with the standard form we find that:

$a=1\\h=-4\\k=-1$

Thus, the vertex of the function is at point (-4,-1)

You might be interested in

how do you find the perpendicular distance between a line segment and a point? (photo attached) i’m stuck

prisoha [69]

Answer:

Exact Distance = $\frac{4\sqrt{10}}{5}$

Approximate Distance = 2.529822

============================================

Explanation:

This process is quite involved if you don't know the shortcut formula. I'll go over the long method first, and then show the shortcut in the next section.

First we'll need the equation of the line through points O and M. Use of the slope formula will show line OM has slope -1/3, which you are correct in stating.

Then use point slope form to determine the equation of line OM is y = (-1/3)x+11/3. This converts to the standard form x+3y = 11.

For anything in the form Ax+By = C, the equation perpendicular to this is of the form Bx-Ay = D. The A,B coefficients swap, and one item is negated. This helps form the negative reciprocal slope needed for the perpendicular line.

Compare x+3y = 11 to Ax+By = C. We see that A = 1 and B = 3.

So Bx-Ay = D turns into 3x-y = D. Then plug in the coordinates of H(-3,2) and compute to get

3x-y = 3(-3)-2 = -9-2 = -11. So D = -11

The equation 3x-y = D turns into 3x-y = -11 which is the equation of the line through point H and this line is perpendicular to line OM.

At this point, we have this system of equations

x+3y = 11

3x-y = -11

Solve that system however you wish. Substitution may be the best choice. Doing so leads to the intersection point (-2.2, 4.4)

The last step is to apply the distance formula between the points H(-3,2) and the intersection point (-2.2, 4.4)

The distance you should get is $\frac{4\sqrt{10}}{5} \approx 2.529822$

I'm skipping steps because listing everything out would take up way too much space in my opinion.

--------------------------------------------------------

The first section goes over a fairly lengthy process of finding the perpendicular distance. Luckily, there's a shortcut.

Consider an equation of the form Ax+By=C, aka standard form. Now consider a point P located at (m,n) that is not on the line Ax+By = C. We can find the distance from P to the line using this formula below

$d = \frac{|Am+Bn-C|}{\sqrt{A^2+B^2}}$

In this case, A = 1, B = 3 and C = 11 found back in the previous section. So you'll still need to calculate the equation of line OM.

Also, we'll use (m,n) = (-3,2) which are the coordinates of point H

From here it's a fairly straightforward computation

$d = \frac{|Am+Bn-C|}{\sqrt{A^2+B^2}}\\\\d = \frac{|1*(-3)+3*2-11|}{\sqrt{(1)^2+(3)^2}}\\\\d = \frac{|-8|}{\sqrt{1+9}}\\\\d = \frac{8}{\sqrt{10}}\\\\$

Optionally we can rationalize the denominator like so

$d = \frac{8}{\sqrt{10}}\\\\d = \frac{8\sqrt{10}}{\sqrt{10}\sqrt{10}}\\\\d = \frac{8\sqrt{10}}{\sqrt{10*10}}\\\\d = \frac{8\sqrt{10}}{\sqrt{100}}\\\\d = \frac{8\sqrt{10}}{10}\\\\d = \frac{2*4\sqrt{10}}{2*5}\\\\d = \frac{4\sqrt{10}}{5}\\\\d \approx 2.529822\\\\$

There are different ways to write down the answer, but they all represent the same number.

4 0

2 years ago

How do I prove that the figure is a rectangle ? <br> I really need help I don’t know what to do.

goldenfox [79]

Answer:

If all angles in a quadrilateral are right angles, then it’s a rectangle (reverse of the rectangle definition). (Actually, you only need to show that three angles are right angles — if they are, the fourth one is automatically a right angle as well.)

If the diagonals of a parallelogram are congruent, then it’s a rectangle (neither the reverse of the definition nor the converse of a property).

If a parallelogram contains a right angle, then it’s a rectangle (neither the reverse of the definition nor the converse of a property).

Tip: Do the following to visualize why this method works: Take an empty cereal box and push in the top flaps. If you then look into the empty box, the top of the box makes a rectangular shape, right? Now, start to crush the top of the box — you know, like you want to make it flat before putting it in the trash. As you start to crush the top of the box, you see a parallelogram shape. Now, after you’ve crushed it a bit, if you take this parallelogram and make one of the angles a right angle, the whole top has to become a rectangle again. You can’t make one of the angles a right angle without the other three also becoming right angles.

Step-by-step explanation:

6 0

3 years ago

Read 2 more answers

Write the following number in scientific notation. 0.0012

Klio2033 [76]

Answer:

1.2 x 10^-3 = 0.0012

Step-by-step explanation:

7 0

3 years ago

A grocery store sells snacks by weight. A 6-oz bag of gummy worms costs $3.60. How much would it cost for a 2-oz bag?