Need help please. Thank you.
1. Determine whether the function has a maximum or minimum. State that value.
f(x)= -2x^2-4x
2. Determine what c would in order to complete the square.
x^2-5x+c
3. Solve the quadratic by factoring.
3x^2-16x+5=0
4. Solve the quadratic by factoring.
x^2+x-48=0
5. Describe the transformation if f(x)=1/3(x+2)^2+5
#1. x^2 has negative coefficient. downward
#2. (5/2)^2
#3. hint: 15+1 = 16
#4. does not factor. Did you mean +2x?
#5. shift left 2, scale vertically by 1/3, shift up 5
Thanks, Steve! Yes on #4 its suppose to be +2x so my bad.
Sure, I'd be happy to help you with your questions! Here are the step-by-step explanations for each question:
1. To determine whether the given function has a maximum or minimum, we can look at the coefficient of the x^2 term. In this case, the coefficient is -2. Since it is negative, we know that the function opens downwards and has a maximum value. To find the value of this maximum, we can use the formula -b/(2a), where a is the coefficient of x^2 (-2 in this case) and b is the coefficient of x (-4 in this case). Plugging in these values, we get x = -(-4) / (2*-2) = -4/(-4) = 1. Therefore, the function has a maximum value of f(1) = -2(1)^2 - 4(1) = -2 - 4 = -6.
2. To complete the square for the quadratic expression x^2 - 5x + c, we need to find the value of c that makes it a perfect square trinomial. We can do this by taking half of the coefficient of x, squaring it, and adding it to both sides of the equation. In this case, the coefficient of x is -5, so we have (-5/2)^2 = 25/4. Adding this to both sides, we get x^2 - 5x + 25/4 = c + 25/4. Therefore, c = 25/4.
3. To solve the quadratic equation 3x^2 - 16x + 5 = 0 by factoring, we need to find two binomials that multiply to give 3x^2 - 16x + 5. The middle term coefficient (-16) can be represented as the sum of two numbers whose product is the product of the leading coefficient (3) and the constant term (5). In this case, these two numbers are -1 and -5 since (-1)*(-5) = 5 and (-1) + (-5) = -6. Therefore, we can rewrite the quadratic as (3x + 1)(x - 5) = 0, which means the solutions are x = -1/3 and x = 5.
4. To solve the quadratic equation x^2 + x - 48 = 0 by factoring, we need to find two binomials that multiply to give x^2 + x - 48. The middle term coefficient (1) can be represented as the sum of two numbers whose product is the product of the leading coefficient (1) and the constant term (-48). In this case, these two numbers are 7 and -6 since 7*(-6) = -42 and 7 - 6 = 1. Therefore, we can rewrite the quadratic as (x + 7)(x - 6) = 0, which means the solutions are x = -7 and x = 6.
5. The given function f(x) = 1/3(x + 2)^2 + 5 is in vertex form, where the vertex is represented by (-2, 5). The transformation of the function can be described as follows:
- The graph is shifted 2 units to the left along the x-axis due to the term (x + 2).
- The graph is stretched vertically by a factor of 1/3 due to the coefficient 1/3.
- The graph is shifted upwards 5 units along the y-axis due to the term +5.
I hope these explanations help! Let me know if there's anything else I can assist you with.