1.Use completing the square to describe the graph of the following function. Support your answer graphically.
f(x) = -2x^2 + 4x + 5
2. Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function. R(x) = 5x/x+7
The vertical asymptote(s) is/are x =
The horizontal asymptote(s) is/are y =
The oblique asymptote(s) is/are y =
f(x) = -2x^2 + 4x + 5
= -2(x^2 - 2x) + 5
= -2(x^2 - 2x + 1) + 5 - -2(1)
= -2(x-1)^2 + 7
this is a parabola with vertex at (1,7), opening downwards, with roots at 1 +/- sqrt(14)/2
5x/(x+7) = 5 - 35/(x+7)
has a vertical asymptote at x=-7
and a horizontal asymptote at y=5
no oblique asymptote, since the degree of top and bottom are the same.
1. To use completing the square to describe the graph of the function f(x) = -2x^2 + 4x + 5, we will follow these steps:
Step 1: Rewrite the function in the standard form of a quadratic equation, which is ax^2 + bx + c.
In this case, -2x^2 + 4x + 5 is already in the standard form.
Step 2: Complete the square by adding and subtracting a constant term inside the parentheses. To determine the constant term, divide the coefficient of x (4) by 2 and square it, which gives (4/2)^2 = 2^2 = 4.
So, we can rewrite the function as:
f(x) = -2(x^2 - 2x + 1) + 5 - (-2)(1)
= -2(x^2 - 2x + 1) + 7
Step 3: Rewrite the quadratic expression inside the parentheses as a perfect square trinomial by factoring or using the formula (a - b)^2 = a^2 - 2ab + b^2. In this case, (x - 1)^2 = x^2 - 2x + 1.
Therefore, we have:
f(x) = -2(x - 1)^2 + 7
Step 4: Determine the vertex of the parabola by using the formula x = -b/2a.
In this case, the coefficient of x^2 is -2, and the coefficient of x is 4.
The vertex x-coordinate = -4 / (2 * -2) = -4 / -4 = 1.
To find the y-coordinate, substitute the x-value back into the original equation:
f(1) = -2(1)^2 + 4(1) + 5 = -2 + 4 + 5 = 7.
Therefore, the vertex of the parabola is (1, 7).
Step 5: Determine the axis of symmetry, which is a vertical line passing through the vertex of the parabola. In this case, the axis of symmetry is x = 1.
Step 6: Use the vertex and the axis of symmetry to draw the graph. The vertex represents the minimum or maximum point of the parabola, and the axis of symmetry divides the parabola into two symmetrical halves.
The graph of the function f(x) = -2x^2 + 4x + 5 is a downward-opening parabola with a vertex at (1, 7) and the axis of symmetry x = 1. The graph will look like a "U" shape, and the point (1, 7) will be at the bottom of the parabola.
2. To find the vertical, horizontal, and oblique asymptotes for the rational function R(x) = 5x / (x+7), we will follow these steps:
Vertical Asymptote:
The vertical asymptote occurs when the denominator of the rational function equals zero. In this case, the denominator is (x+7). Therefore, we set (x+7) = 0 and solve for x.
By solving (x+7) = 0, we get x = -7.
Hence, the vertical asymptote is x = -7.
Horizontal Asymptote:
To find the horizontal asymptote, we need to consider the degrees of the polynomials in the numerator and denominator.
In this case, the degree of the numerator is 1 and the degree of the denominator is also 1. Since the degrees are equal, we divide the leading coefficients of the polynomials.
The leading coefficient of the numerator is 5 and the leading coefficient of the denominator is 1.
Hence, the horizontal asymptote is y = 5 / 1 = 5.
Oblique Asymptote:
To find the oblique asymptote, we need to check if the degrees of the numerator and denominator differ by 1.
In this case, the degree of the numerator is 1 and the degree of the denominator is also 1. Therefore, there is no oblique asymptote.
In summary, for the rational function R(x) = 5x / (x+7):
- The vertical asymptote(s) is/are x = -7.
- The horizontal asymptote(s) is/are y = 5.
- There is no oblique asymptote.