What is the equation in slope-intercept form of the line that passes through the points 0, negative 4 and ?
{silence}Option A.{silence}
y equals 2 x plus 2
{silence}Option Be.{silence}
y equals 1 over 2 x minus 4
{silence}Option C.{silence}
y equals 2 x minus 4
{silence}Option D.{silence}
y equals 1 over 2 x plus 2
The slope-intercept form of a line is given by y = mx + b, where m represents the slope of the line and b is the y-intercept.
We know the line passes through the point (0, -4), which must be the y-intercept since the x value is 0. This means b = -4.
However, the second point through which the line passes was not provided in the question. Without knowing the second point, we cannot determine the slope (m) of the line.
Given that b = -4, we can eliminate Option A and Option D because they do not have the correct y-intercept.
Between the two remaining options:
Option B: y = (1/2)x - 4
Option C: y = 2x - 4
Without the second point to determine the slope, we cannot choose between these options. Please provide the second point so I can help you find the correct slope and the equation of the line.
A rectangle has a length of x cubed y to the fourth inches and a width of x y to the seventh inches. Which expression represents the ratio of the length of the rectangle to the width of the rectangle?
{silence}Option A.{silence}
x squared over y cubed
{silence}Option Be.{silence}
x cubed over y cubed
{silence}Option C.{silence}
y cubed over x cubed
{silence}Option D.{silence}
y cubed over x squared
To find the ratio of the length to the width of the rectangle, you should divide the length by the width.
Given that:
Length = \( x^3 y^4 \)
Width = \( xy^7 \)
The ratio would be:
Length / Width = \( \frac{x^3 y^4}{xy^7} \)
When dividing exponents with the same base, you subtract the exponents:
\( \frac{x^3}{x} = x^{3-1} = x^2 \)
\( \frac{y^4}{y^7} = y^{4-7} = y^{-3} \) which is also equal to \( \frac{1}{y^3} \)
So the ratio of length to width simplifies to:
\( x^2 y^{-3} \) or \( \frac{x^2}{y^3} \)
Therefore, the correct option is:
{silence}Option A.{silence}
x squared over y cubed
If the solutions of p of x equals 0 are negative 14 and 11, which function could be p?
If the solutions (roots) of the polynomial \( p(x) = 0 \) are \( x = -14 \) and \( x = 11 \), then the polynomial can be expressed by the factors that correspond to these roots. Each root \( x = a \) corresponds to a factor of the form \( (x - a) \).
Therefore, for the root \( x = -14 \), the corresponding factor is \( (x - (-14)) \) or \( (x + 14) \), and for the root \( x = 11 \), the corresponding factor is \( (x - 11) \).
Multiplying these two factors together will give us a polynomial \( p(x) \) that has \( x = -14 \) and \( x = 11 \) as solutions:
\[ p(x) = (x + 14)(x - 11) \]
Expanding this, we get:
\[ p(x) = x^2 - 11x + 14x - 154 \]
\[ p(x) = x^2 + 3x - 154 \]
Therefore, the function \( p \) could be:
\[ p(x) = x^2 + 3x - 154 \]
A teacher’s age is 6 years greater than 2 times a student’s age. A principal’s age is 10 years greater than 3 times the student’s age.
If x represents the student’s age in years, which expression represents how many years older the principal is than the teacher?
Let's start by turning the words into algebraic expressions based on the value of x, which represents the student's age.
Teacher's age: 6 years greater than 2 times the student's age can be represented as:
Teacher's age = 2x + 6
Principal's age: 10 years greater than 3 times the student's age can be represented as:
Principal's age = 3x + 10
Now, to find the expression for how many years older the principal is than the teacher, we subtract the teacher's age from the principal's age:
Principal's age - Teacher's age = (3x + 10) - (2x + 6)
Simplify the expression by combining like terms:
= 3x + 10 - 2x - 6
= (3x - 2x) + (10 - 6)
= x + 4
Thus, the expression that represents how many years older the principal is than the teacher is x + 4 years.
Quadratic functions q and w are graphed on the same coordinate grid.
The vertex of the graph of q is 18 units below the vertex of the graph of w.
Which pair of functions could have been used to create the graphs of q and w?
To answer the question about which pair of quadratic functions could create graphs where the vertex of q is 18 units below the vertex of w, we need to consider the general vertex form of a quadratic function, which is:
\[ f(x) = a(x - h)^2 + k \]
Here, (h, k) is the vertex of the parabola. Given the information, we could create a myriad of pairs of quadratic functions with respective vertices (h, k) for w and (h, k-18) for q, assuming that the "18 units below" refers to the k (or y) values of their vertices.
However, without specific options or additional information on the functions' leading coefficients (a) or the x-coordinate of the vertex (h), we cannot precisely determine which pair of functions were used.
Please provide the specific functions or further details so that we can find an accurate pair of functions q and w.
A gardener measured the heights of two plants at the end of every week. The function y equals 3 x plus 8.5 gives the height of Plant A in centimeters at the end of x weeks. The function y equals 2.5 x plus 14.5 gives the height of Plant B in centimeters at the end of x weeks.
Based on this information, which of these statements is true?