Write the following quadratic function in vertex form. State the x and y-intercepts and the range. 𝑦 = −4𝑥^2 + 20𝑥 − 17

Answer 1

To convert the quadratic function 𝑦 = −4𝑥^2 + 20𝑥 − 17 into vertex form, we need to complete the square.

First, let's factor out the common factor of -4 from the first two terms:
𝑦 = -4(𝑥^2 - 5𝑥) - 17

Next, we will complete the square by taking half of the coefficient of 𝑥 (-5), squaring it and adding it inside the parentheses:
𝑦 = -4(𝑥^2 - 5𝑥 + 6.25) - 17 - 4(6.25)

Now, let's simplify:
𝑦 = -4(𝑥 - 2.5)^2 - 17 - 25

Combining like terms, we have:
𝑦 = -4(𝑥 - 2.5)^2 - 42

Therefore, the quadratic function 𝑦 = −4𝑥^2 + 20𝑥 − 17 can be written in vertex form as 𝑦 = -4(𝑥 - 2.5)^2 - 42.

To find the x-intercepts, we set 𝑦 equal to zero and solve the resulting quadratic equation:
0 = -4(𝑥 - 2.5)^2 - 42
4(𝑥 - 2.5)^2 = -42
(𝑥 - 2.5)^2 = -10.5
Since the square of any real number is always greater than or equal to zero, there are no x-intercepts for this quadratic function.

The y-intercept occurs when 𝑥 is equal to zero. Plugging 𝑥 = 0 into the quadratic function, we get:
𝑦 = -4(0)^2 + 20(0) - 17
𝑦 = -17
Therefore, the y-intercept is -17.

To determine the range of the function, let's analyze the coefficient of the squared term (-4). Since the coefficient is negative, the parabola opens downward. This means that the y-values will be negative or zero for all possible x-values. Hence, the range of the function is (-∞, 0].

Answer 2

wrong again, on the x-intercepts!
y = −4𝑥^2 + 20𝑥 − 17 = -(2x-5)^2 + 8
so it opens downward, with vertex at (5/2,8)
the x-intercepts, using the quadratic formula,are
x = 5/2 ± √2

Answer 3

You are correct, I apologize for the mistake.

To find the x-intercepts, we can set the function equal to zero and solve for x:
0 = -4𝑥^2 + 20𝑥 - 17

Using the quadratic formula, x = (-b ± √(b^2 - 4ac))/(2a),
we have a = -4, b = 20, and c = -17.

Plugging in these values, we get:
x = (-20 ± √(20^2 - 4(-4)(-17)))/(2(-4))
x = (-20 ± √(400 - 272))/(-8)
x = (-20 ± √128)/(-8)
x = (-20 ± √(64*2))/(-8)
x = (-20 ± 8√2)/(-8)
x = 5/2 ± √2

Therefore, the x-intercepts are x = 5/2 + √2 and x = 5/2 - √2.

Thank you for catching that mistake!