f(x)=7(x–4)2+3.
To find the vertex form of the quadratic function f(x) = 7(x-4)^2 + 3, we need to expand the square:
f(x) = 7(x^2 - 8x + 16) + 3
Next, distribute the 7:
f(x) = 7x^2 - 56x + 112 + 3
Simplify:
f(x) = 7x^2 - 56x + 115
Therefore, the quadratic function f(x) = 7(x-4)^2 + 3 is equivalent to the standard form f(x) = 7x^2 - 56x + 115.
The given expression is a quadratic function in the form of f(x) = a(x-h)^2 + k, where 'a' represents the coefficient of the quadratic term, (h, k) represents the vertex of the parabola, and 'x' is the independent variable.
In this case, the function is f(x) = 7(x-4)^2 + 3, where 'a' is 7, (h, k) is (4, 3), and 'x' is our input variable.
To find the vertex of the parabola, we can directly read it from the expression. The vertex of the parabola is given by (h, k), which in this case, is (4, 3). So, the vertex is located at the point (4, 3) on the coordinate plane.
To analyze the shape of the parabola, we look at the coefficient 'a'. Since 'a' is positive (7), the parabola opens upwards.
To find the x-intercepts (zeros) of the function, we set f(x) = 0 and solve for x.
0 = 7(x-4)^2 + 3
-7(x-4)^2 = 3
(x-4)^2 = -3/7
x - 4 = ±√(-3/7)
x = 4 ± √(-3/7)
Since the expression inside the square root is negative, there are no real solutions for x. Therefore, the given function does not have any x-intercepts.
To find the y-intercept of the function, we substitute x = 0 into the expression.
f(0) = 7(0-4)^2 + 3
f(0) = 7(-4)^2 + 3
f(0) = 7(16) + 3
f(0) = 112 + 3
f(0) = 115
Therefore, the y-intercept of the function is 115, which means the parabola intersects the y-axis at the point (0, 115).
In summary, the given function f(x) = 7(x-4)^2 + 3 represents a parabola that opens upwards with a vertex at (4, 3), no x-intercepts, and a y-intercept at (0, 115).
To understand the function f(x)=7(x-4)^2+3, we can break it down into several components:
1. The expression '(x-4)^2' represents a quadratic function. It means that we take the value of 'x', subtract 4 from it, and square the result.
2. The quadratic expression '(x-4)^2' is then multiplied by the coefficient 7. This means that the whole quadratic expression is scaled by a factor of 7.
3. Finally, the result of step 2 is added to 3. This means that we take the value from step 2 and add 3 to it.
In summary, the function f(x)=7(x-4)^2+3 is a quadratic function with a vertex at (4, 3) and a vertical stretch by a factor of 7.
The price of an item was lowered by 25%. By what percent must this discounted price be increased to go back to original price?
responses
25%
30%
33%
40%
To go back to the original price after a 25% discount, we need to increase the discounted price by a certain percentage.
Let's say the original price is x. After a 25% discount, the price becomes 0.75x (since 0.75 is 100% - 25%).
To find the percentage increase to go back to the original price, we need to calculate the difference between the original price and the discounted price, and then express that difference as a percentage of the discounted price.
Difference = x - 0.75x = 0.25x
To find the percentage increase, we divide the difference by the discounted price and multiply by 100:
Percentage increase = (0.25x / 0.75x) * 100 = (1/3) * 100 = 33.33...
Therefore, the discounted price must be increased by approximately 33.33% to go back to the original price. So the correct answer is 33%.