Answer:
[tex]\boxed{\sf x = 1, \ x = -5}[/tex]
Explanation:
[tex]\rightarrow \sf 4x^2 + 16x - 20 = 0[/tex]
[tex]\rightarrow \sf 4(x^2 + 4x - 5) = 0[/tex]
[tex]\rightarrow \sf x^2 + 4x - 5 = 0[/tex]
[tex]\rightarrow \sf x^2 + 5x -x- 5 = 0[/tex]
[tex]\rightarrow \sf x(x + 5) -1(x+ 5) = 0[/tex]
[tex]\rightarrow \sf (x-1)(x+ 5) = 0[/tex]
[tex]\rightarrow \sf x = 1, \ x = -5[/tex]
Find an equation in standard form for the hyperbola with vertices at (0, ±9) and foci at (0, ±10)
Therefore, the equation of the hyperbola with a given origin has vertices at (0, ±9) and foci at (0, ±10) is [tex]\frac{x^{2} }{81}-\frac{y^{2} }{19} =1[/tex].
Given, that a hyperbola centred at the origin has vertices at (0, ±9) and foci at (0, ±10).
What is hyperbola?In mathematics, a hyperbola is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows.
The formula for a hyperbola centred at the origin is [tex]\frac{(x-h)^{2} }{a^{2} } -\frac{(y-k)^{2} }{b^{2} } =1[/tex]
Where (h, k) is the center = (0, 0)
Distance from centre to vertices a = 9 ⇒ a² = 81
Distance from centre to vertices which is given from the foci c = 10
⇒ c² = 100
Using the Pythagorean formula, c²= a²+ b²
Substituting the values 100 = 81 + b²
So we get, b²= 100 - 81 = 19
Substituting the values in the standard form [tex]\frac{x^{2} }{81}-\frac{y^{2} }{19} =1[/tex].
Therefore, the equation of the hyperbola with a given origin has vertices at (0, ±9) and foci at (0, ±10) is [tex]\frac{x^{2} }{81}-\frac{y^{2} }{19} =1[/tex].
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Quadratic Functions graph
Answer:
y=-x^2-4x-1
Step-by-step explanation:
when it is an arch like an upside down U then its
-x^2
it passes through the y axis at -1
its vertex or high point is -2,3 so its
-4x
Answer:
See below
Step-by-step explanation:
Dome shaped parabola so 'a' is negative
'c' is determined where x = 0
this value is -1
Axis of symmetry is the x value of the vertex (which is the high point of the dome)
x = -2
PLEASE HELP!!
Discussion Topic
Earlier in this course, you explored Euclidean geometry, which is the study of flat space. This
approach follows the teachings of Euclid, in which he describes the relationships between
points, lines, and planes without any numerical measurement. You saw evidence of Euclidean
geometry inside several proofs and geometric constructions.
In contrast, the focus of this unit is understanding geometry using positions of points in a
Cartesian coordinate system. The study of the relationship between algebra and geometry was
pioneered by the French mathematician and philosopher René Descartes. In fact, the
Cartesian coordinate system is named after him. The study of geometry that uses coordinates
in this manner is called analytical geometry.
It's clear that this course teaches a combination of analytical and Euclidean geometry. Based
on your experiences so far, which approach to geometry do you prefer? Why? Which
approach is easier to extend beyond two dimensions? What are some situations in which one
approach to geometry would prove more beneficial Han the other? Describe the situation
and why you think analytical or Euclidean geometry is more applicable.
Based on my experiences so far, the approach to geometry that I prefer is: Euclidean Geometry. This is because the problems are easy to visualize since they are restricted to two-dimensional planes.
Which approach is easier to extend beyond two dimensions?The approach that is easier to extend beyond two dimensions is Euclidean Geometry. Again, this is because of how it deals with shapes and visualization of the same.
Take for instance a triangle; it is easy to go from a two-dimensional equilateral triangle to a square pyramid.
What are some situations in which one approach to geometry would prove more beneficial than the other?Analytical geometry is a superior technique for discovering objects (points, curves, and planes) based on their qualities in some situations than Euclidean geometry is in others (for example, when employing topography or building charts).
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What is the value of n in the equation 2n² + 4n = 0? What is the value of n in the equation 2n² + 4n = 0 ?
N = 0
2(0)² + 4(0) = 0
0 + 0 = 0
A chord AB divides a circle of radius 8 cm into two segments. If AB subtends a
central angle of 45, find the area of the minor segment.
Answer:
Area of the minor segment = 2.505324230749 cm²
Step-by-step explanation:
Area of the minor segment
= Area of the sector ABC - Area of the triangle ABC
……………………………………………………………………………………
• Area of the sector ABC
= Area of the whole circle ÷ 8
= (π×8²) ÷ 8
= 8π
______
•• Area of the triangle ABC
= (1÷2) × CA × CB × sin(45)
= (1÷2) × 8 × 8 × (√2/2)
= 16√2
______
Thus
Area of the minor segment
= 8π - 16√2
= 2.505324230749
How many Identities are there in algebra?
Name them.
Answer:
the technical number depends on what you are including to be algebraic identities.
There are multiple forms of most identities (pos. / neg. and multiple ways of writing the same identity)
generally, you can consider there to be 10
Step-by-step explanation:
algebraic identity: an equality that holds true for any variable values
standard identities:
square of binomial:
(a+b)² = a² +2ab + b²
(a - b)² = a² - 2ab + b²
difference of squares:
(a + b)(a - b) = a² - b²
----
product of two binomials:
(x + a)(x + b) = x² + (a + b)x + ab
square of trinomial:
(x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx
(x - y - z)² = x² + y² + z² -2(xy + yz - zx)
cube of binomial:
(x + y)³ = x³ + y³ + 3xy (x + y)
(x - y)³ = x³ + y³ - 3xy (x - y)
(a - b)³ = a³ - 3a²b + 3ab² - b³
sum of cubes:
a³ + b³ = (a + b)(a² - ab + b²)
x³ + y³ = (x + y)(x² - xy + y²)
x³ + y³ = (x - y)(x² + xy + y²)
difference of cubes:
a³ - b³ = (a - b)(a² + ab + b²)
x³ - y³ = (x - y)³ + 3xy(x - y)
--
x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx)
= 1/2 (x + y + z) [(x -y)² + (y - z)² + (z - x)²]
(a + b + c)³ = a³ + b³ + c³ + 3(a + b)(b + c)(c + a)
--
and
if x + y + z = 0, then x³ + y³ + z³ = 3xyz
hope this helps!!
(this took me a while to write; there are also a few complicated identities also that I've left out)
[tex]( {a + b})^{2} = {a}^{2} + {b}^{2} + 2ab \\ \\ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab \\ \\ {a}^{2} - {b}^{2} = (a + b)(a - b) \\ \\ {a + b + c}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2ab + 2bc + 2ca \\ \\ {a + b}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b) \\ \\ {a - b}^{3} = {a}^{3} - {b}^{3} - 3ab(a - b) \\ \\ {a}^{3} + {b}^{3} + {c}^{3} - 3abc = (a + b + c)( {a}^{2} + {b}^{2} + {c}^{2} - ab - bc - ca).[/tex]
What is the equation of the line that has a slope of -4 and passes through the point (2,3)?
Answer:
The answer is y = -4x+11
Write the ratio as a fraction in lowest terms.
Compare in inches.
4 feet to 49 inches
Answer:
48/49
Step-by-step explanation:
divison and multiplcation
Name the property that justifies the statment If AB - BC = 12, then AB = 12 + BC.
The property that justifies the statement is the commutative law of addition
Commutative law of additionThis law occurs if the same number or constant is added to both sides of an equation.
Given the equation
AB - BC = 12
Add BC to both sides
AB - BC + BC = 12 + Bc
AB = 12 + BC
Hence the property that justifies the statement is the commutative law of addition
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If the sides of a square are increased by 2 inches, the area becomes 25 squares inches. Find the length of sides of the original square.
You know that the sides of a square are increased by 2 inches: the expression should be written as (x+2)
The area is 25 square inches
The area of a square is s×s or s²
so it can be solved as:
(x+2)²=25
[tex](\sqrt{x+2})^{2} =\sqrt{25}[/tex]
x+2=5
x=3
The length of the original side is 3 inches
Hope it helps!
Please helpppp!!!!!!!!!!
The vertex form of the quadratic function f(x) = x² + 2x + 3 is given as follows:
f(x) = (x + 1)² + 2
What is the equation of a parabola given it’s vertex?The equation of a quadratic function, of vertex (h,k), is given by:
y = a(x - h)² + k
In which a is the leading coefficient.
In this problem, the function is:
f(x) = x² + 2x + 3
Completing the squares, we find the equation in vertex form, as follows:
f(x) = (x + 1)² + 2
As (x + 1)² = x² + 2x + 1, hence 2 has to be added to find the equivalent function.
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What’s the slope of this graph? Can y’all guys help please ! Thank y’all
Answer:
-3
Step-by-step explanation:
/= negative slope; \= positive slope
The y-intercept is 2 because this is where it crosses the y-axis
Then you find the next point that touches the graph on an exact point which is (-2,-4)
So, your points are (0,2) and (-2,-4) then you do the slope formula of y2-y1 divided by x2-x1
-4-2/-2-0 = 3 and because the line is / then it is -3
Consider the given density curve.
A density curve is at y = one-third and goes from 3 to 6.
What is the value of the median?
3
4
4.5
6
Picture posted below
Answer: 4.5
Step-by-step explanation:
[tex]\frac{6+3}{2}=\boxed{4.5}[/tex]
We want to find the median for the given density curve.
The value of the median is 4.5. ({3+6)/2}
Let's see how to solve this.
First, for a regular set {x₁, ..., xₙ} we define the median as the middle value.
The difference between a set and a density curve is that the density curve is continuous, so getting the exact middle value can be harder.
Here, we have a constant density curve that goes from 3 to 6.
Because it is constant, the median will just be equal to the mean, thus the median is the average between the two extreme values.
Remember that the average between two numbers a and b is given by:(a + b)/2
So we get: m = (6 + 3)/2 = 4.5
So we can conclude that the value of the median is 4.5, so the correct option is the second one, counting from the top.
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what is 4³ writ
ten in expanded form?
Answer:
4^3 written in expanded form is 4*4*4
Mark as Brainliest please
Answer: 64
Step-by-step explanation:
Hii!
Do you need to know what 4^3 written in expanded form is? No problem!
4^3=4*4*4 (the ^3 tells us to multiply the base (which is 4 here) times 3)
The answer is 64.
Hope that this helped! Best wishes.
--
The function f(x) = ex-3 when evaluated for f(4) is equivalent to?
Answer:
[tex]f(4)=e^{4}-3[/tex] or [tex]f(4)\approx51.5981500331...[/tex]
Step-by-step explanation:
The following is a general explanation of functions, what they mean, and how to evaluate them. The simplifications and calculations at the end are based on the clarification from your comments that [tex]f(x)=e^{x}-3[/tex] . If [tex]f(x)[/tex] is equal to something else, the simplifications/calculation will obviously be a little different, and the answer will likely be different, but the general process will be the same.
Understanding function notation
In the equation [tex]f(x)=e^{x}-3[/tex] , the left side of the equation is saying several things at once.
Names
The first character, "f", is the name of the function.
The name of a function tells you which set of directions you should look for to apply to an input. Some functions have names that are more than one letter, and even might give a hint as to what they do, for instance I might name a certain function [tex]squareroot[/tex], and based on the name, you might guess what the equation for [tex]squareroot(x)[/tex] might look like.
Mathematicians often use "f" as a function name between different problems over and over and over again, unless they come up with a special function that is worth giving a special name to.
Input
The rest of the characters after the name of the function, are "(x)" which is a pair of parentheses to contain the input, and the input itself.
Again, mathematicians often use "x" as a generic input over and over and over again, until we actually know which number we want to put into the function.
Output
Even though we've already discussed every character in the notation on the left, there is actually one more thing that the left side of the equation is saying: Altogether, "f(x)" is the output of the function, when the input "x" is put in. I feel it's worth stating, because it's often glossed over.
The equation itself
Next in the equation is the equals sign, which means that the left side of the equation, which is the output "f(x)", is equal to the stuff on the right.
The right side of the equation is the rule to follow describing what f(x) does to an input, and can help you calculate the output.
For instance, if I asked what p(4) was, it is clear that there is an input of 4, but without knowing what the function "p" does, there's no way to know what the function will give as an output when one inputs "4". This is why the right-hand side of the equation for a function is important.
Evaluating functions
To evaluate any function, we take an input, apply the function, and simplify/calculate the output.
In this case, the question asks for [tex]f(4)[/tex], meaning that the "input" is "4".
In this case, the question gives an explanation of what the function does through a rule given in equation form.
Given your clarification of the given function, [tex]f(x)=e^{x}-3[/tex] , meaning a generic input (in this case "x") is put into the function "f" and the output is found by doing the arithmetic on the right side of the equation (exponentiating the number by "e", and then subtracting 4)
Notice that in the expression [tex]f(4)[/tex], the "x" was replaced with "4". So, we need to substitute "4" where we see an "x" in the rule on the right hand side of the equation rule, and simplify and/or calculate (calculate, like with a calculator)
So, to evaluate the expression [tex]f(4)[/tex], simply substitute, and simplify/calculate.
Simplifying (no calculator)
To simplify without a calculator, we substitute "4". Any time you make a substitution, use parentheses. Then, try to simplify.
[tex]f(4)=e^{(4)}-3[/tex]
[tex]f(4)=e^{4}-3[/tex]
As it turns out, this is as much as we could simplify things here (without a calculator). So, if an exact answer is needed, this would be it.
Calculating (with a calculator)
To calculate with a calculator, we substitute "4". Any time you make a substitution, use parentheses. Then, try to simplify.
[tex]f(4)=e^{(4)}-3[/tex]
[tex]f(4)=e^{4}-3[/tex]
[tex]f(4)=54.5981500331...-3[/tex]
[tex]f(4)=51.5981500331...[/tex]
So calculating with a calculator, the output of the function "f", for an input of "4", is approximately 51.598.
What are the solution(s) to the quadratic equation x² - 25 = 0?
h
O x=5 and x =
-5
−5
x = 25 and x = -25
x = 125 and x = -125
Ono real solution’s
Answer:
x = ±5
Step-by-step explanation:
x² - 25 = 0
x² = 25
√x² = √25
x = ±5
Check:
5² - 25 = 0
-5² - 25 = 0
25 - 25 = 0
what are x and 8 in the linear equation? 8/x+5=50/25
A.One is variable and one is a coefficient
B.One is a constant and one is a coefficient?
C.Both are variables.
D.both are constants.
The graphs of three functions are shown. Which statements accurately compare the functions on the graph? Select two options. The square root and quadratic function share a y-intercept. The square root and absolute value function share an x-intercept. The range of the square root and quadratic function are the same. The range of the square root and absolute value function are the same. The quadratic function and the absolute value function have a minimum while the square root function has a maximum.
The following statements accurately compare the three functions on the graph: option A and C.
How to interpret the graph?In Geometry, the following statements are true about the graph of a function:
The x-intercepts of two or more functions are the same when their curves meet the x-axis at the same point. The y-intercepts of two or more functions are the same when their curves meet the y-axis at the same point. The ranges of two or more functions are the same when they all have the same vertical extension. The absolute value function and square root function both have the same minimum i.e at lower end of their range.By critically observing the graph of a function, we can logically deduce that the following statements accurately compare the three functions on the graph:
"The square root and quadratic function share a y-intercept.""The range of the square root and absolute value function are the same."Read more on function here: brainly.com/question/4246058
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Analyze the conditional statement below and complete the instructions that follow.
If an angle measures 90°, then it is a right angle.
Identify ~q.
O An angle does not have measure 90°.
OIt is not a right angle.
An angle measures 90°.
OIt is a right angle.
Answer: It is not a right angle.
Terry wants to pour cement around the edge of the circular patio in her backyard.The patio has a radius of 5 feet.What is the distance,in feet,around the edge of the patio?Use 3.14 for Pi.PLEASE EXPLAIN
E.15.7
F.31.4
G.49.3
H.78.5
Answer:
F. 31.4 ft
Step-by-step explanation:
The distance around the edge of a circle is called the circumference.
Formula for the circumference of a circle
[tex]\sf Circumference\:of\:a\:circle=2 \pi r[/tex]
(where r is the radius)
Given values:
radius = 5 ftπ = 3.14Substitute the given values into the formula:
[tex]\begin{aligned}\implies \sf circumference & = \sf 2 \cdot 3.14 \cdot 5\\ & = \sf 31.4\:ft \end{aligned}[/tex]
Therefore, the distance around the edge of the circular patio is 31.4 ft.
The distance around the edge of the patio is 31.4 feet, which is determined by the circumference of a circle. The correct answer is option (F).
To find the distance around the edge of the circular patio, we need to calculate the circumference of the circle.
The formula for the circumference of a circle is given by:
Circumference = 2 × π × radius
Given that the radius of the patio is 5 feet and using the approximate value of π as 3.14, we can now calculate the circumference:
Circumference = 2 × 3.14 × 5
Circumference = 31.4 feet
Therefore, the distance around the edge of the patio is 31.4 feet.
Hence, the correct answer is option (F).
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What does the ratio 4: 18 represent in this situation?
Answer:
2:9 or 2/9
Step-by-step explanation:
I guess this question asks the simplest form of ratio 4:18
so I divided it by two and since they cannot be divided further that's the answer
you can either choose to write it in ratio form or fraction form that's your choice good day.
If f(x) = 5x - 1, then f-1 (x)
Step-by-step explanation:
[tex]f(x) = 5x - 1[/tex]
[tex] \: [/tex]
[tex]y = 5x - 1[/tex]
[tex]5y - 1 = x[/tex]
[tex]5y = x + 1[/tex]
[tex]y = \frac{x + 1}{5} [/tex]
[tex] {f}^{ - 1} (x) = \frac{x + 1}{5} [/tex]
QUESTION 8 - 1 POINT
The median monthly rent in 1994 was $499 and $633 in 2001. What was the percent increase in the
median monthly rent prices from 1994 to 2001? Round to the nearest tenth of a percent.
Answer:
26.85%
Step-by-step explanation:
first you find the difference in price increase
633-499=134
*percentage increase=increase value/original value x100
so;=134/499x100=28.85%
hope it helps you.please don't forget to give me a thanks
Which of the following statements must be true about this diagram? Check
all that apply.
A. mz2+ mz3 = mz4
B. mz1+ m2 2 = m/3
C. m23 is greater than m<2
D. m24 is greater than mz1
E. mz1+mz2 = mz4
F. mz4 is greater than m<2
Answer:
D, E, F is the answer.
Step-by-step explanation:
From the diagram:-
m∠ 4 is greater than m∠ 1m∠ 4 is greater than m∠ 2m∠ 1 + m∠ 2 = m∠ 4Therefore, D, E, F.
-------------------------------
what is the area of a triangle (-4,-5),(3,-3),(3,-7)?
i will give brainliest please help quick
The area of a triangle with vertices (-4,-5), (3,-3), and (3,-7) will be 7 square units.
What is the area of the triangle?The vertices of the triangle are (x₁, y₁), (x₂, y₂), and (x₃, y₃).
Then the area of the triangle will be
Area = 1/2 |[(x₁y₂ + x₂y₃ + x₃y₁) - (y₁x₂ + y₂x₃ + y₃x₁)]|
We have
(x₁, y₁) ⇒ (-4, -5)
(x₂, y₂) ⇒ (3, -3)
(x₃, y₃) ⇒ (3, -7)
Then the area will be
Area = 1/2[{(-4) x (-3) + (3) x (-7) + (3) x (-5)} – {(-5) x (3) + (-3) x (3) + (-7) x (-4)}]
Area = 1/2 [(12 – 9 – 21) – (- 15 – 9 + 28)]
Area = 1/2[-18 + 4]
Area = 1/2 × 14
Area = 7 square units
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Given functions f ( x ) = 1 √ x and m ( x ) = x 2 − 4 , state the domains of the following functions using interval notation.
Domain of f ( x ) /m ( x ):
Domain of f ( m ( x ) ):
Domain of m(f(x)):
The domain of a function f(x)/m(x) = 1/√x(x² - 4) is (0, ∞) - {0, 2, -2} for other function is shown in the solution.
What is a function?It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
We have:
f(x) = 1/√x
m(x) = x² - 4
Domain of f(x)/m(x):
f(x)/m(x) = (1/√x)/(x² - 4)
f(x)/m(x) = 1/√x(x² - 4)
The denominator cannot be zero:
√x(x² - 4) ≠ 0
x(x - 2)(x+2) ≠ 0
x ≠ 0, 2, -2
and x > 0
Domain of f(x)/m(x) is: (0, ∞) - {0, 2, -2} or [tex]\:\left(0,\:2\right)\cup \left(2,\:\infty \:\right)[/tex]
Domain of f(m(x)):
f(m(x)) = 1/√(x² - 4)
x² - 4 > 0
Domain: [tex]\rm \left(-\infty \:,\:-2\right)\cup \left(2,\:\infty \:\right)[/tex]
Domain of m(f(x)):
= ((1/√x)² - 4)
Domain: [tex]\:\left(0,\:\infty \:\right)[/tex]
Thus, the domain of a function f(x)/m(x) = 1/√x(x² - 4) is (0, ∞) - {0, 2, -2} for other function is shown in the solution.
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A quadrilateral has vertices E(–4, 2), F(4, 7), G(8, 1), and H(0, –4). Which statements are true? Check all that apply.
A) The slope of EH is -8/5
B) The slopes of EF and GH are both 5/8
C) FG is perpendicular to GH.
D) Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
E) Quadrilateral EFGH is a rectangle because all angles are right angles.
Answer:
D) Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
B) The slopes of EF and GH are both 5/8
Step-by-step explanation:
The following inference can be made about each answer choices:
For option A
A) The slope of EH is 3/2 and not -8/5
For EH, slope = (2--4)/(0--4) = 6/4 =3/2
For option B
B) The slopes of EF and GH are both 5/8
slope of a graph can be determined by change in y axis/ change in x axis
For EF, slope = (7-2)/(4--4) = 5/8
for GH, slope = (1--4)/(8-0) = 5/8
For option C
C) FG is perpendicular to GH, its can't be determined since it is concluded that the quadrilateral is a parallelogram, therefore this option is wrong.
For option D
D) Quadrilateral EFGH is a parallelogram because both pairs of opposite sides are parallel.
Slope of a graph can be determined by rise divided by run change in y axis/ change in x axis
For EF, slope = (7-2)/(4--4) = 5/8
for GH, slope = (1--4)/(8-0) = 5/8
For EH, slope = (2--4)/(0--4) = 6/4 =3/2
for FG, slope = (7-1)/(8-4) = 6/4 =3/2
EF and GH give the same slope, likewise EH and FG and they are opposite to each other, this make them parallel. Therefore, EFGH is a parallelogram.
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Need help!!!!!!!!!!!!!!!!!
Write an equation in point-slope form of the line that passes through the given point and with the given slope m
(-9,2); m=6
The equation is [tex]\boxed{y-2=6(x+9)}[/tex].
Use a calculator to graph the functions y=-3x-21 and y=x³-3x-3. How many points of intersection do the
functions share?
0
1
2
3
Answer:
1
Step-by-step explanation:
When you graph the functions they intersect at one point
(-3.368119, -31.10436)