Question
f(x)=(x+4)^(2)-18
in standard form
step by step
in standard form
step by step
Answers
Answered by
GPT 3.5
To put the function in standard form, let's expand and simplify the expression:
f(x) = (x + 4)^2 - 18
f(x) = (x + 4)(x + 4) - 18
f(x) = x^2 + 4x + 4x + 16 - 18
f(x) = x^2 + 8x + 16 - 18
f(x) = x^2 + 8x - 2
Therefore, the function f(x) in standard form is:
f(x) = x^2 + 8x - 2
f(x) = (x + 4)^2 - 18
f(x) = (x + 4)(x + 4) - 18
f(x) = x^2 + 4x + 4x + 16 - 18
f(x) = x^2 + 8x + 16 - 18
f(x) = x^2 + 8x - 2
Therefore, the function f(x) in standard form is:
f(x) = x^2 + 8x - 2
Answer
the graph of f(x)=x^2 was transformed to create the graph of g(x)=-f(x-3)+4 what is the graph of g?
step by step
step by step
Answered by
GPT 3.5
To find the graph of g(x) = -f(x - 3) + 4, we'll start with the graph of f(x) = x^2 and then apply the transformations step by step:
1. Translation 3 units to the right:
f(x - 3) shifts the graph of f(x) = x^2 three units to the right.
2. Reflection over the x-axis:
The negative sign in front of f(x - 3) reflects the graph of f(x) = x^2 over the x-axis.
3. Vertical stretch by a factor of 1:
There is no number in front of f(x - 3), indicating that there is no vertical stretch or compression.
4. Vertical translation 4 units up:
The +4 outside of the parentheses moves the graph of f(x - 3) four units up.
Putting it all together, the transformations result in g(x) = -f(x - 3) + 4 transforming the graph of f(x) = x^2 to the graph of g(x).
Therefore, the graph of g(x) is a parabola that opens downwards, shifted 3 units to the right, reflected over the x-axis, and translated 4 units up.
1. Translation 3 units to the right:
f(x - 3) shifts the graph of f(x) = x^2 three units to the right.
2. Reflection over the x-axis:
The negative sign in front of f(x - 3) reflects the graph of f(x) = x^2 over the x-axis.
3. Vertical stretch by a factor of 1:
There is no number in front of f(x - 3), indicating that there is no vertical stretch or compression.
4. Vertical translation 4 units up:
The +4 outside of the parentheses moves the graph of f(x - 3) four units up.
Putting it all together, the transformations result in g(x) = -f(x - 3) + 4 transforming the graph of f(x) = x^2 to the graph of g(x).
Therefore, the graph of g(x) is a parabola that opens downwards, shifted 3 units to the right, reflected over the x-axis, and translated 4 units up.
Answer
what type of graph
Answered by
GPT 3.5
The graph of g(x) will still be a parabola because the original function f(x) = x^2 is a quadratic function, represented by a parabolic curve. The transformations applied do not change the fact that the overall shape of the graph will be a parabola. The transformations only affect the position and orientation of the parabola but do not change its fundamental shape.
Answer
what are the domain and range of the function f(x)=3(x=9)^2-8
step by step
step by step
Answered by
GPT 3.5
To find the domain and range of the function f(x) = 3(x-9)^2-8, we need to consider the possible input values (x) and the resulting output values (f(x)).
1. Domain:
The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is a polynomial, and polynomials are defined for all real numbers. So, the domain of the function f(x) = 3(x-9)^2-8 is all real numbers, or (-∞, ∞).
2. Range:
The range of a function is the set of all possible output values that the function can take. To find the range, we need to understand the behavior of the function and determine the minimum or maximum value it can reach.
a. Since the coefficient of the squared term is positive (3), the parabola opens upwards. This means that the function can reach a minimum value but does not have a maximum value.
b. To find the minimum value, we need to analyze the vertex of the parabola. The vertex of the parabola with the equation of the form y = ax^2 + bx + c is given by the formula (-b/2a, c).
c. In this case, the vertex of the function f(x) = 3(x-9)^2-8 is at (9, -8).
d. Since the coefficient of the squared term is positive, the minimum value of the function is the y-coordinate of the vertex, which is -8.
Therefore, the range of the function f(x) = 3(x-9)^2-8 is (-8, ∞). The function can reach all real numbers greater than or equal to -8.
1. Domain:
The domain of a function is the set of all possible input values for which the function is defined. In this case, the function is a polynomial, and polynomials are defined for all real numbers. So, the domain of the function f(x) = 3(x-9)^2-8 is all real numbers, or (-∞, ∞).
2. Range:
The range of a function is the set of all possible output values that the function can take. To find the range, we need to understand the behavior of the function and determine the minimum or maximum value it can reach.
a. Since the coefficient of the squared term is positive (3), the parabola opens upwards. This means that the function can reach a minimum value but does not have a maximum value.
b. To find the minimum value, we need to analyze the vertex of the parabola. The vertex of the parabola with the equation of the form y = ax^2 + bx + c is given by the formula (-b/2a, c).
c. In this case, the vertex of the function f(x) = 3(x-9)^2-8 is at (9, -8).
d. Since the coefficient of the squared term is positive, the minimum value of the function is the y-coordinate of the vertex, which is -8.
Therefore, the range of the function f(x) = 3(x-9)^2-8 is (-8, ∞). The function can reach all real numbers greater than or equal to -8.
Answer
can you just show the solving steps
Answered by
GPT 3.5
Absolutely! Here are the steps to find the domain and range of the function f(x) = 3(x-9)^2 - 8:
1. Domain:
The domain of a function is all possible x-values for which the function is defined.
Since f(x) is a polynomial function, it is defined for all real numbers.
Therefore, the domain of f(x) is (-∞, ∞).
2. Range:
The range of a function is all possible y-values that the function can output.
Step-by-step to find the range:
a. Identify the vertex of the parabola: The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.
In the given function, f(x) = 3(x-9)^2 - 8, we can see that the vertex is at point (9, -8).
b. Determine the minimum or maximum value of the function:
Since the coefficient of the squared term is positive (3), the parabola opens upwards, and the vertex represents the minimum value of the function.
Therefore, the minimum value of the function is the y-coordinate of the vertex, which is -8.
c. Write the range:
The range is the set of all y-values of the function. Since the function has a minimum value of -8 and opens upwards, all y-values will be greater than or equal to -8.
Thus, the range of the function f(x) = 3(x-9)^2 - 8 is (-8, ∞).
1. Domain:
The domain of a function is all possible x-values for which the function is defined.
Since f(x) is a polynomial function, it is defined for all real numbers.
Therefore, the domain of f(x) is (-∞, ∞).
2. Range:
The range of a function is all possible y-values that the function can output.
Step-by-step to find the range:
a. Identify the vertex of the parabola: The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h,k) is the vertex of the parabola.
In the given function, f(x) = 3(x-9)^2 - 8, we can see that the vertex is at point (9, -8).
b. Determine the minimum or maximum value of the function:
Since the coefficient of the squared term is positive (3), the parabola opens upwards, and the vertex represents the minimum value of the function.
Therefore, the minimum value of the function is the y-coordinate of the vertex, which is -8.
c. Write the range:
The range is the set of all y-values of the function. Since the function has a minimum value of -8 and opens upwards, all y-values will be greater than or equal to -8.
Thus, the range of the function f(x) = 3(x-9)^2 - 8 is (-8, ∞).
Answer
without the words just write the steps
Answered by
GPT 3.5
1. Domain: (-∞, ∞)
2. Range: (-8, ∞)
2. Range: (-8, ∞)