1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
neonofarm [45]
2 years ago
13

2y+3x² +5+y+2x+x²+2 what are the coefficients?

Mathematics
1 answer:
saw5 [17]2 years ago
8 0

Answer: 200

The quadratic function f(x) = a(x - h)2 + k, a not equal to zero, is said to be in standard form. If a is positive, the graph opens upward, and if a is negative, then it opens downward. The line of symmetry is the vertical line x = h, and the vertex is the point (h,k).

Any quadratic function can be rewritten in standard form by completing the square. (See the section on solving equations algebraically to review completing the square.) The steps that we use in this section for completing the square will look a little different, because our chief goal here is not solving an equation.

Note that when a quadratic function is in standard form it is also easy to find its zeros by the square root principle.

Example 3.

Write the function f(x) = x2 - 6x + 7 in standard form. Sketch the graph of f and find its zeros and vertex.

f(x) = x2 - 6x + 7.

= (x2 - 6x )+ 7.        Group the x2 and x terms and then complete the square on these terms.

= (x2 - 6x + 9 - 9) + 7.

We need to add 9 because it is the square of one half the coefficient of x, (-6/2)2 = 9. When we were solving an equation we simply added 9 to both sides of the equation. In this setting we add and subtract 9 so that we do not change the function.

= (x2 - 6x + 9) - 9 + 7. We see that x2 - 6x + 9 is a perfect square, namely (x - 3)2.

f(x) = (x - 3)2 - 2. This is standard form.

From this result, one easily finds the vertex of the graph of f is (3, -2).

To find the zeros of f, we set f equal to 0 and solve for x.

(x - 3)2 - 2 = 0.

(x - 3)2 = 2.

(x - 3) = ± sqrt(2).

x = 3 ± sqrt(2).

To sketch the graph of f we shift the graph of y = x2 three units to the right and two units down.

If the coefficient of x2 is not 1, then we must factor this coefficient from the x2 and x terms before proceeding.

Example 4.

Write f(x) = -2x2 + 2x + 3 in standard form and find the vertex of the graph of f.

f(x) = -2x2 + 2x + 3.

= (-2x2 + 2x) + 3.

= -2(x2 - x) + 3.

= -2(x2 - x + 1/4 - 1/4) + 3.

We add and subtract 1/4, because (-1/2)2 = 1/4, and -1 is the coefficient of x.

= -2(x2 - x + 1/4) -2(-1/4) + 3.

Note that everything in the parentheses is multiplied by -2, so when we remove -1/4 from the parentheses, we must multiply it by -2.

= -2(x - 1/2)2 + 1/2 + 3.

= -2(x - 1/2)2 + 7/2.

The vertex is the point (1/2, 7/2). Since the graph opens downward (-2 < 0), the vertex is the highest point on the graph.

Exercise 2:

Write f(x) = 3x2 + 12x + 8 in standard form. Sketch the graph of f ,find its vertex, and find the zeros of f. Answer

Alternate method of finding the vertex

In some cases completing the square is not the easiest way to find the vertex of a parabola. If the graph of a quadratic function has two x-intercepts, then the line of symmetry is the vertical line through the midpoint of the x-intercepts.

The x-intercepts of the graph above are at -5 and 3. The line of symmetry goes through -1, which is the average of -5 and 3. (-5 + 3)/2 = -2/2 = -1. Once we know that the line of symmetry is x = -1, then we know the first coordinate of the vertex is -1. The second coordinate of the vertex can be found by evaluating the function at x = -1.

Example 5.

Find the vertex of the graph of f(x) = (x + 9)(x - 5).

Since the formula for f is factored, it is easy to find the zeros: -9 and 5.

The average of the zeros is (-9 + 5)/2 = -4/2 = -2. So, the line of symmetry is x = -2 and the first coordinate of the vertex is -2.

The second coordinate of the vertex is f(-2) = (-2 + 9)(-2 - 5) = 7*(-7) = -49.

Therefore, the vertex of the graph of f is (-2, -49).

You might be interested in
Round 227033 to the nearest ten thousand
Sonbull [250]

I a pretty sure it is 228,000 but i am not sure.

7 0
3 years ago
Read 2 more answers
Hi, I need the are to this question, I just need to show my work btw​
Nana76 [90]
What is the question?
6 0
3 years ago
4/5 divided by 1/10?
Evgesh-ka [11]

Answer:

the answer to the question is 8

6 0
2 years ago
Read 2 more answers
If 3 - 3k-1 = 81, what is the value of k?
andreyandreev [35.5K]
Answer: (C) 4

Have a good day.
8 0
2 years ago
Read 2 more answers
Joe decides to take his six year old son to the planetarium. The price for the child's ticket is 5.75 dollars less than the pric
mart [117]
To answer this item, we are instructed that the price of the ticket for the children is equal to x. Since, we are also given that this is 5.75 less than the price of tickets for the adults, we may express this as follows,

    x = n - 5.75

n is the price of ticket for the adults. This can be calculated by transposing 5.75 to the other side of the equation,

   n = x + 5.75

The answer is the third choice. 
6 0
2 years ago
Other questions:
  • Q7 Q19.) Find the area of the triangle having the given measurements.  
    5·1 answer
  • . A circular window has a diameter of 40 centimeters. What is the circumference of the window? Use 3.14 for π π 251.2 cm
    5·1 answer
  • Select the correct answer.
    15·1 answer
  • What is the height of a right circular cylinder made from a rectangular sheet of lengil
    8·1 answer
  • Is the equation 8x12x5 =- 8x(12x5) an associative property or distributive property equation?
    14·2 answers
  • What is the volume of the solid shown above
    15·1 answer
  • At what point does the graph of 3x + 4y = 15 intersect the graph of x^2 + y^2 = 9? Express any non-integer coordinate as a commo
    10·2 answers
  • How much interest will be earned on a $1600 investment at 7.72% compounded
    5·1 answer
  • Please help!<br><br> Thank you!
    13·1 answer
  • Larry has a bag of candy that weighs 12 pounds. He wants to divide the candy into 5 equal groups. How much will each group weigh
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!