determine the y-intercept of f(x)=0.2(x-5)^2 - 8, then graph this function on the grid provided. Label any points you plot with the ordered pair.
To determine the y-intercept of the function f(x), we substitute x = 0 into the function.
f(x) = 0.2(x-5)^2 - 8
f(0) = 0.2(0-5)^2 - 8
= 0.2(-5)^2 - 8
= 0.2(25) - 8
= 5 - 8
= -3
Therefore, the y-intercept of f(x) is -3.
To graph this function:
1. Set up a coordinate grid.
2. Plot the y-intercept, which is the point (0, -3).
3. Find other points by plugging in different x-values into the equation and calculating their corresponding y-values.
4. Connect the plotted points to form a smooth curve.
The graph should show a downward-opening parabola since the coefficient of the x^2 term is positive.
what is the equation for f(x)=0.2(x-5)^2 - 8
The equation for the function f(x) = 0.2(x-5)^2 - 8 is already provided.
determine the y-intercept of f(x)=0.2(x-5)^2 - 8
To determine the y-intercept of the function f(x) = 0.2(x-5)^2 - 8, we substitute x = 0 into the equation:
f(x) = 0.2(x-5)^2 - 8
f(0) = 0.2(0-5)^2 - 8
= 0.2(-5)^2 - 8
= 0.2(25) - 8
= 5 - 8
= -3
The y-intercept of the function f(x) = 0.2(x-5)^2 - 8 is -3.
equation of axis of symmetry of f(x)=0.2(x-5)^2 - 8
The equation for the axis of symmetry of a quadratic function in the form f(x) = a(x-h)^2 + k is given by x = h.
In this case, the equation f(x) = 0.2(x-5)^2 - 8 is already in the vertex form, where the vertex is located at the point (h, k). Therefore, the axis of symmetry is the vertical line x = 5.
So, the equation of the axis of symmetry of the function f(x) = 0.2(x-5)^2 - 8 is x = 5.
vertex of f(x)=0.2(x-5)^2 - 8
The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
In the given function f(x) = 0.2(x-5)^2 - 8, we can identify that h = 5 and k = -8.
Therefore, the vertex of the function f(x) = 0.2(x-5)^2 - 8 is (5, -8).
when Jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by a quadratic function. the important parts like vertex, x-intercept, y-intercept, axis of symmetry.
To represent the flight of the golf ball with a quadratic function, we can use the vertex form: f(x) = a(x-h)^2 + k. In this case, the variables a, h, and k will represent specific values related to the flight of the ball.
Based on the given information:
- The vertex of the quadratic function represents the highest or lowest point of the ball's flight. We are given that the height of the ball is 21m at a horizontal distance of 91m. Therefore, the vertex of the quadratic function is (91, 21).
- The x-intercept(s) of the quadratic function represents the point(s) where the ball hits the ground. We know that the ball traveled a total horizontal distance of 130m, so the x-intercept(s) will be the points where x = 0 and x = 130.
- The y-intercept of the quadratic function represents the point where the ball was hit or its initial height. We are not given the initial height in this case, so we cannot determine the y-intercept.
- The axis of symmetry of the quadratic function is a vertical line that passes through the vertex. The equation of the line is given by x = h. In this case, the axis of symmetry would be x = 91.
To determine the x-intercept(s) of the parabolic function, we need to solve the equation f(x) = 0.
In this case, the equation would be: 0 = a(x-h)^2 + k
Since the ball hits the ground at x = 0 and x = 130, we can substitute those values into the equation to find the value(s) of a:
When x = 0:
0 = a(0 - 91)^2 + 21
0 = a(-91)^2 + 21
0 = 8281a + 21
When x = 130:
0 = a(130 - 91)^2 + 21
0 = a(39)^2 + 21
0 = 1521a + 21
Solving these equations will allow us to determine the value of a and thus, the x-intercept(s) of the quadratic function.
when Jaxon hit a golf ball off the grass with a 9-iron it travelled 130m horizontally until it landed back on the grass. the height of the ball was measured to be 21m when it reached a horizontal distance of 91m. the flight of the golf ball can be represented by a quadratic function.
To represent the flight of the golf ball with a quadratic function, we can use the vertex form: f(x) = a(x-h)^2 + k. The variables a, h, and k represent specific values related to the flight of the ball.
Given:
- The ball traveled a horizontal distance of 130m.
- The height of the ball was 21m when it reached a horizontal distance of 91m.
To find the equation of the quadratic function, we need to find the vertex (h, k) and the value of a.
The vertex, (h, k), represents the highest or lowest point of the ball's flight. We know that the height of the ball was 21m when it reached a horizontal distance of 91m. Therefore, the vertex of the quadratic function is (91, 21).
Now, let's find the value of a.
We can use the vertex form of the quadratic equation to find the value of a. Plugging in the given vertex (91, 21) and a point on the graph (91, 21), we have:
21 = a(91 - 91)^2 + k
21 = a(0)^2 + k
21 = a * 0 + k
21 = k
We also need to find the value of a. To find a, we can use another point given in the problem. We are told that the ball traveled a total horizontal distance of 130m. When the ball lands back on the grass, the height is 0. Plugging in (130, 0) into the vertex form, we have:
0 = a(130 - 91)^2 + k
0 = a(39)^2 + 21
0 = 1521a + 21
Solving the equation 0 = 1521a + 21 for a, we find that a = -21/1521 = -1/73.
Therefore, the equation of the quadratic function representing the flight of the golf ball can be written as:
f(x) = (-1/73)(x - 91)^2 + 21
skye solved a quadratic equation and said the solutions of that equation are -5 and 4. Oscar believes that the equation skye solved was (x - 5)(x + 4) = 0.
- verify skye's roots in Oscar's equation
- is Oscar's belief correct?
- If Oscar is incorrect, provide a revised equation that would work for
skye's solutions.
To verify Skye's roots (-5 and 4) in Oscar's equation (x - 5)(x + 4) = 0, we substitute these values into the equation:
For x = -5:
(-5 - 5)(-5 + 4) = (-10)(-1) = 10
For x = 4:
(4 - 5)(4 + 4) = (-1)(8) = -8
Since the results are not zero, Skye's roots (-5 and 4) do not satisfy Oscar's equation (x - 5)(x + 4) = 0.
Therefore, Oscar's belief is incorrect.
To find a revised equation that would work for Skye's solutions, we can use the factored form of a quadratic equation:
(x - a)(x - b) = 0, where a and b represent the roots.
For Skye's roots (-5 and 4), the revised equation would be:
(x + 5)(x - 4) = 0
Determine the exact roots of 11x^2 - 0.25 = -3x using the quadratic formula.
To determine the exact roots of the quadratic equation 11x^2 - 0.25 = -3x using the quadratic formula, we first need to rewrite the equation in standard form:
11x^2 + 3x - 0.25 = 0
Comparing this equation to the general quadratic equation ax^2 + bx + c = 0, we can identify that a = 11, b = 3, and c = -0.25.
The quadratic formula states that the roots of the equation ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c:
x = (-(3) ± √((3)^2 - 4(11)(-0.25))) / (2(11))
Simplifying:
x = (-3 ± √(9 + 11)) / 22
x = (-3 ± √20) / 22
Further simplifying:
x = (-3 ± 2√5) / 22
Therefore, the exact roots of the equation 11x^2 - 0.25 = -3x are:
x = (-3 + 2√5) / 22
x = (-3 - 2√5) / 22
the sweet stuff food truck sells various sweet treats. their to selling treat is the deluxe cinnamon roll. they sell an average of 24 deluxe cinnamon rolls on a typical summer evening at $6.00 per roll. According to market research for every $0.25 decrease in price, the food truck can expect to sell 2 more cinnamon rolls each night.
- according to these numbers, determine the current revenue from the
deluxe cinnamon.
The current revenue from the deluxe cinnamon rolls can be calculated by multiplying the average number of rolls sold by the price per roll.
Given:
- Average number of deluxe cinnamon rolls sold on a typical summer evening: 24
- Price per roll: $6.00
Revenue = Average number of rolls sold * Price per roll
Revenue = 24 * $6.00
Revenue = $144.00
Therefore, the current revenue from the deluxe cinnamon rolls is $144.00.
the sweet stuff food truck sells various sweet treats. their to selling treat is the deluxe cinnamon roll. they sell an average of 24 deluxe cinnamon rolls on a typical summer evening at $6.00 per roll. According to market research for every $0.25 decrease in price, the food truck can expect to sell 2 more cinnamon rolls each night.
- define variables and set up a function to model the scenario. [hint: take a look at example 2 in lesson 2.4 if you need help defining variables and setting up a function]
Let's define the following variables:
- x: the number of $0.25 decreases in price
- y: the increase in the number of cinnamon rolls sold due to each $0.25 decrease in price
Based on the given information, for every $0.25 decrease in price, the food truck can expect to sell 2 more cinnamon rolls each night. This means that the increase in the number of cinnamon rolls sold (y) is directly proportional to the number of $0.25 decreases in price (x), and we can write it as y = kx, where k is the constant of proportionality.
We know that when there are no $0.25 decreases in price (x = 0), the number of cinnamon rolls sold is the average of 24 rolls (y = 24).
So, we have the following data points:
(x, y) = (0, 24)
Now, to find the value of k, we can use another data point. We are told that for every $0.25 decrease in price, the food truck can expect to sell 2 more cinnamon rolls. This can be expressed as (x, y) = (1, 26).
Using these two data points, we can set up a system of equations to solve for k:
24 = k(0)
26 = k(1)
From the first equation, we find that k = 24/0, which is undefined. Therefore, we cannot find a value for k using this equation.
We can look at the second equation, where k = 26/1 = 26.
So, the function that models the scenario is:
y = 26x
This equation represents the increase in the number of cinnamon rolls sold (y) for each $0.25 decrease in price (x).
the sweet stuff food truck sells various sweet treats. their to selling treat is the deluxe cinnamon roll. they sell an average of 24 deluxe cinnamon rolls on a typical summer evening at $6.00 per roll. According to market research for every $0.25 decrease in price, the food truck can expect to sell 2 more cinnamon rolls each night.
- determine the maximum revenue from the deluxe cinnamon rolls that the food truck can earn. what will be the price of a deluxe cinnamon roll when the revenue is at a maximum?
To determine the maximum revenue from the deluxe cinnamon rolls, we need to consider the relationship between the price of the rolls and the number of rolls sold.
According to the information given, for every $0.25 decrease in price, the food truck can expect to sell 2 more cinnamon rolls. This implies that as the price decreases, the number of rolls sold increases.
To find the maximum revenue, we need to find the price at which the revenue is maximized. Revenue is calculated by multiplying the number of rolls sold by the price per roll.
Let's denote the decrease in price in terms of $0.25 as x and the corresponding increase in the number of rolls sold as 2x. Referencing the original price of $6.00, the new price per roll will be $6.00 - $0.25x and the corresponding number of rolls sold will be 24 + 2x.
Now, we can calculate the revenue:
Revenue = Price per roll * Number of rolls sold
R = (6.00 - 0.25x)(24 + 2x)
To find the maximum revenue, we need to find the value of x that corresponds to the maximum value of R by using optimization techniques or by examining the graph of the function.
However, in this case, based on the information given, the relationship between the price and the number of rolls sold is linear. As the price decreases, the number of rolls sold increases in a predictable manner. Consequently, the revenue function R(x) will be maximized when the price per roll is at its lowest point.
Here, the lowest price per roll corresponds to the maximum revenue. Since the price decreases by $0.25, we need to find the maximum number of decreases that can occur while keeping the price positive (i.e., not lower than $0).
Let's set up an inequality:
6.00 - 0.25x > 0
Solving this inequality, we find:
-0.25x > -6.00
x < 24
The number of $0.25 decreases cannot exceed 24 since we cannot have a negative price. However, note that 24 decreases would bring the price to $0.
Therefore, the maximum number of $0.25 decreases is 23, which occurs when the price is $6.00 - (23 * $0.25) = $0.25.
Hence, the price of a deluxe cinnamon roll when the revenue is at a maximum is $0.25.
describe how you can use the characteristics of quadratic functions(i.e. not a table of values) to determine the ordered pairs of 5 distinct points on the quadratic function with the following characteristics:
- zero at x = -4
- axis of symmetry at x = 2
- max value of 9
- y-intercept at 8
*you may find it helpful to create a sketch to go along with your description.
To determine the ordered pairs of 5 distinct points on the quadratic function with the given characteristics, we can use the general form of a quadratic function: f(x) = a(x-h)^2 + k.
1. Zero at x = -4:
Since the function has a zero at x = -4, it means that (x+4) is a factor of the quadratic equation. Therefore, (x+4) = 0, which yields x = -4 as one of the x-intercepts. This gives us the ordered pair (-4, 0).
2. Axis of symmetry at x = 2:
The axis of symmetry in a quadratic function is given by the equation x = h, where h is the x-coordinate of the vertex. In this case, the axis of symmetry is x = 2. Therefore, one point on the graph is (2, y), where y can vary depending on the specific function.
3. Maximum value of 9:
The maximum value occurs at the vertex of the parabola. The y-coordinate of the vertex is the maximum value of the quadratic function. In this case, the maximum value is 9
describe how you can use the characteristics of quadratic functions(i.e. not a table of values) to determine the ordered pairs of 5 distinct points on the quadratic function with the following characteristics:
- zero at x = -4
- axis of symmetry at x = 2
- max value of 9
- y-intercept at 8
describe how you can use the characteristics of quadratic functions to determine the ordered pairs of 5 distinct points on the quadratic function with the following characteristics:
- zero at x = -4
- axis of symmetry at x = 2
- max value of 9
- y-intercept at 8