Hey thanks for all your help but you kinda confused me on a few:
1)Determine whether f(x)=-5x^2-10x+6 has a maximum or minimum value and find that value
A.minimum -1
B.maximum 11
C.maximum -1
D.minimum 11
and you said the function opens up, so there is a minimum.
it occurs when x=-1, and that minimum is f(-1), which is 11. So the answer is D right?
2)Identify the vertex,axis of symmetry,and direction of opening for y=1/2(x-8)^2+2
A.(8,2);x=-8;up
B.(-8,-2);x=-8;down
C.(8,-2);x=8;up
D.(8,2);x=8;up
I picked A,here is your reasoning.vertex is ok, but how can the axis of symmetry be x=-8?? Would it not go through the vertex?? So it is x=9 (always the same as the x of the vertex)
3)Which quadratic function has its vertex at(-2,7)and opens down?
A.y=-3(x+2)^2+7
B.y=(x-2)^2+7
C.y=-12(x+2)^2-7
D.y=-2(x-2)+7
4)Write y=x^2+4x-1 in vertex form.
A.y=(x-2)^2+5
B.y=(x+2)^2-5
C.y=(x+2)^2-1
D.y=(x+2)^2+3
Well I'm not Reiny but I can help you out with #3
I found out by plugging it into the Y=
for creating graphs on my TI-83 Plus that it is A.
You should be able to do the same if you have this calculator and plug in each function. Then hit graph
For #1 contrary to what you found I found that when I plugged it into my Y= function on my calculator that the
-curve faced down
-at X=-1 was the vertex
-Y=10.9/ 11
- based on that I would have to say my conclusion is that it has a maximum at Y=11
For #2 it is D
but not x=9 like you said but I assume it was a typo
The axis of symetry is the same as the x of the vertex (your thinking is correct)
1. yes it is D
2. I clearly meant to type x=8
3. clearly A. B would have vertex (2,7), and both C and D open downwards
4. y=x^2+4x-1
y = x^2 + 4x + 4 - 4 - 1
= (x+2)^2 - 5
1) To determine whether the function f(x) = -5x^2 - 10x + 6 has a maximum or minimum value, we need to analyze the coefficient of the x^2 term. Since the coefficient is negative (-5), the parabola opens downward, which means it has a maximum value.
To find the x-coordinate of the vertex (where the maximum/minimum occurs), we can use the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms, respectively. In this case, a = -5 and b = -10.
Using the formula, we have x = -(-10)/(2(-5)) = 10/(-10) = -1.
So, the parabola has a maximum value when x = -1.
To find the corresponding y-value (the maximum value), we substitute x = -1 into the function: f(-1) = -5(-1)^2 - 10(-1) + 6 = -5 + 10 + 6 = 11.
Therefore, the maximum value of the function is 11.
The correct answer is D) minimum 11.
2) For the quadratic function y = 1/2(x-8)^2 + 2, let's identify its vertex, axis of symmetry, and direction of opening.
The vertex form of a quadratic function is given by y = a(x-h)^2 + k, where (h, k) represents the vertex of the parabola.
Comparing it with our given function y = 1/2(x-8)^2 + 2, we can see that the vertex is located at (8, 2).
The axis of symmetry is a vertical line that passes through the vertex. So, the correct answer is a line that passes through the x-coordinate of the vertex.
Thus, the correct answer is B) (-8, -2); x = -8.
Regarding your question about the axis of symmetry going through the vertex, you are correct. The axis of symmetry is always a vertical line passing through the vertex.
3) To find a quadratic function with a vertex at (-2, 7) and opens down, we need to use the vertex form of a quadratic function: y = a(x-h)^2 + k.
Comparing it with the given vertex (-2, 7), we have h = -2 and k = 7.
Since the function opens down, the coefficient 'a' must be negative.
By substituting the values of h, k, and the direction of opening into the vertex form, we get:
y = -a(x-(-2))^2 + 7
y = -a(x+2)^2 + 7
So, the correct answer is C) y = -12(x+2)^2 - 7.
4) To write the quadratic function y = x^2 + 4x - 1 in vertex form, we need to complete the square.
First, let's group the terms with x: y = (x^2 + 4x) - 1.
Next, we need to add and subtract the square of half the coefficient of x. The coefficient of x is 4, so half of it is 2. The square of 2 is 4.
y = (x^2 + 4x + 4 - 4) - 1.
We can rewrite it as:
y = ((x^2 + 4x + 4) - 4) - 1.
Now, we can factor the perfect square trinomial inside the parentheses:
y = ((x + 2)^2 - 4) - 1.
Simplifying further, we have:
y = (x + 2)^2 - 4 - 1.
y = (x + 2)^2 - 5.
The correct answer is A) y = (x - 2)^2 + 5.
To check your answers, you can use the Y= function on your TI-83 Plus calculator to graph each function and observe the vertex, direction of opening, and axis of symmetry.