Some Cool Math Games
Q1. what is the radius of a circle whose equation is x2+y2+8x−6y+21=0? 2 units 3 units 4 units 5 units.
To find the radius of a circle given its equation, we need to rewrite the equation in the standard form:
(x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius.
Let's rewrite the equation x^2 + y^2 + 8x - 6y + 21 = 0:
x^2 + 8x + y^2 - 6y + 21 = 0
To complete the square for x terms, we add (8/2)^2 = 16 to both sides:
x^2 + 8x + 16 + y^2 - 6y + 21 = 16
Now, let's complete the square for y terms by adding (6/2)^2 = 9 to both sides:
x^2 + 8x + 16 + y^2 - 6y + 9 + 21 = 16 + 9
Simplifying further:
(x^2 + 8x + 16) + (y^2 - 6y + 9) = 25
(x + 4)^2 + (y - 3)^2 = 25
Comparing this to the standard form, we can see that the center of the circle is (-4, 3) and the radius is √25 = 5 units.
Therefore, the radius of the circle is 5 units.
Q2. which is the graph of f(x) = 100(0.7)x?
The graph of the function f(x) = 100(0.7)^x is an exponential decay function.
In an exponential decay function, the base value is between 0 and 1, which causes the function to decrease as x increases.
Here's a rough description of the graph:
As x approaches negative infinity, f(x) approaches 0 but never reaches it.
As x approaches positive infinity, f(x) approaches 0 but from the positive side.
The graph is always above the x-axis, as the exponential function never produces negative values.
The graph starts high on the y-axis (at f(0) = 100) and decreases exponentially as x increases. However, because the base is less than 1, the function decays quickly. The rate of decay is determined by the value of the base (0.7 in this case).
To see the exact shape and details of the graph, it would be best to use graphing software or a graphing calculator.
Q3. which graph shows the axis of symmetry for the function f(x) = (x – 2)2 + 1?
The function f(x) = (x - 2)^2 + 1 represents a quadratic function in vertex form. The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
In the given function f(x) = (x - 2)^2 + 1, the vertex is located at the point (2, 1). The axis of symmetry for the graph is a vertical line passing through the x-coordinate of the vertex, which in this case is x = 2.To identify the graph that shows the axis of symmetry, we need to look for the vertical line labeled x = 2. The graph that has this vertical line passing through x = 2 represents the axis of symmetry for the function.
Unfortunately, since this is a text-based format and I cannot visually display the graphs, I cannot directly show you which graph represents the axis of symmetry. However, if you have a set of graphs or images representing different functions, you can look for the one that includes a vertical line at x = 2, and that will correspond to the axis of symmetry for the function f(x) = (x - 2)^2 + 1.
Q4. For what values of m does the graph of y = 3x2 + 7x + m have two x-intercepts?
To determine the values of m for which the graph of the quadratic function y = 3x^2 + 7x + m has two x-intercepts, we need to examine the discriminant of the quadratic equation.
The quadratic equation is in the form ax^2 + bx + c = 0, where a = 3, b = 7, and c = m. The discriminant (denoted as Δ) is given by the formula Δ = b^2 - 4ac.
For the quadratic function to have two x-intercepts, the discriminant must be greater than zero (Δ > 0). This indicates that there are two distinct real solutions for x.
Let's apply this criterion to our equation:
Δ = (7)^2 - 4(3)(m)
Δ = 49 - 12m
For two x-intercepts, Δ > 0. Therefore:
49 - 12m > 0
Now, let's solve this inequality for m:
49 > 12m
49/12 > m
Simplifying the expression, we find:
m < 4.08333...
Therefore, the values of m that make the quadratic function y = 3x^2 + 7x + m have two x-intercepts are m < 4.08333... (rounded to a reasonable decimal approximation). In interval notation, this can be written as (-∞, 4.08333...).
Q5. How to solve y=mx+b
The equation y = mx + b represents a linear equation in slope-intercept form. To solve this equation, we are typically looking to find the value of y when given specific values for x, or finding the values of x and y that satisfy the equation.
Here are the main steps to solve a linear equation in slope-intercept form:
Given x or y, substitute the known values into the equation.
Solve for the unknown variable.
If necessary, simplify the equation or rearrange it to isolate the unknown variable.
Verify the solution by substituting it back into the original equation.
Let's look at a few examples to illustrate the process:
Example 1: Find y when x = 2 in the equation y = 3x + 4.
Substitute x = 2 into the equation: y = 3(2) + 4.
Simplify: y = 6 + 4.
Calculate: y = 10.
The solution is y = 10.
Example 2: Find x when y = -5 in the equation y = 2x - 3.
Substitute y = -5 into the equation: -5 = 2x - 3.
Add 3 to both sides to isolate 2x: -5 + 3 = 2x.
Simplify: -2 = 2x.
Divide both sides by 2: -1 = x.
The solution is x = -1.
Remember, if you are given specific values for x and y, you can substitute them into the equation to find the value of the other variable. If you need to find a specific variable, you can isolate it by simplifying and rearranging the equation.
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