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Question

1. Find all points where the graphs of x^2-Y=3 and y=x-1 intersect 


2. Write the equation of the line that goes through the point (0,3) And is parallel to the line 2x+5y=10 .write your answer in standard form 

3.Solve the equation over [0,2pi] 
2Cosx=(sqr root)3 

4. Is the function y=2x^2 +3 odd even or neither? Explain your choice

Answer 1

2).

The standard form line equation is Ax + By = C. A shouldn't be negative, A and B shouldn't both be zero, and A, B and C should be integers.

The line equation is 2x + 5y = 10.

Write the equation in slope - intercept form.

5y = - 2x - 10

y = (- 2/5)x - 2.

Comapare the equation with slope - intercept form.

Slope (m) = - 2/5.

Because the parallel lines have same slopes, the slope of parallel line through the point (0, 3) is - 2/5.

Now the parallel line equation is y = (- 2/5)x + b.

Find the y - intercept by substituting the point in the parallel line equation say (x, y) = (0, 3).

3  = (- 2/5)(0) + b

b = 3 - 0

⇒ b = 3.

The parallel line equation is y = (- 2/5)x + 3.

5y = - 2x + 15

Add 2x to each side.

2x + 5y = 15.

The standard form the equation of the parallel line is 2x + 5y = 15.

3).

The trigonometric equation is 2cos (x) = √3.

Divide each sid eby 2.

cos (x) = √3/2.

cos (x) = cos(π/6)

The genaral solution of cos(θ) = cos(α) is θ = 2nπ ± α, where n is an integer.

⇒ x = 2nπ ± π/6

If n = 0, x = 2(0)π + (π/6) and x = 2(0)π - (π/6) = π/6 and - π/6.

If n = 1, x = 2(1)π + (π/6) and x = 2(1)π - (π/6) = 2π + π/6 and 2π - π/6 = 13π/6 and 11π/6,

Therefore, the solutions of the given equation are x = π/6 and x = 11π/6 in the interval [0, 2π].

Answer 2

1).

The equations are x² - y = 3 and y = x - 1.

Substitute y = x - 1 in x² - y = 3.

x² - (x - 1) = 3

x² - x + 1 - 3 = 0

x² - x - 2 = 0

By factoring by grouping.

x² - 2x + x - 2 = 0

x(x - 2) + 1(x - 2) = 0

(x - 2)(x + 1) = 0

x = 2  and  x = - 1.

If x = 2, then y = 2 - 1 = 1.

If x = - 1, then y = - 1 - 1 = - 2.

Therefore, the intersection points are (2, 1) and (- 1, - 2).

4).

Let f (x) be the real value function.

Remember that :

• Even : f (- x) = f (x).
• Odd : f (- x) = - f (x).

The function f (x) = 2x² + 3.

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

So, the function is even.