What are the vertex, axis of symmetry, maximum or
minimum value, and range of y=-5(x+2)^2-8?
please explain to me how you would solve this..I don't really understand.
The answer is given..
The vertex is (-2, -8);
the axis of symmetry is
x= -2; k= -8 is the
maximum value; the domain
is all real numbers; the range
is all real numbers less than or equal to -8.
Well, isn't that a quadratic equation that looks like it's ready to party! Let's break it down step-by-step.
First, let's identify the equation: y = -5(x+2)^2 - 8. This is in what we call vertex form, which is super convenient for us.
- The vertex form of a quadratic equation is y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex. In our case, the vertex is (-2, -8). So, the coordinates of the vertex have been revealed, just like pulling a rabbit out of a hat!
- The axis of symmetry is like a magical line that cuts the quadratic equation right down the middle, creating symmetry on both sides. In this case, the equation is centered around x = -2 (the x-coordinate of the vertex). This is where the clown car of symmetry rides!
- The maximum or minimum value is determined by the coefficient "a" of the equation. Since we have -5 in front of the (x+2)^2 term, it means our graph is facing downwards like a sad clown frown. The height of this frown determines the maximum value. In this case, the maximum value is -8, just like a balloon that slipped out of your hand and floated away.
- Now, for the range and the domain. The domain of this quadratic equation is ALL real numbers because you can plug in any real number for "x" and calculate a corresponding "y". It's like juggling infinite possibilities, except with numbers instead of balls.
- As for the range, it's all real numbers less than or equal to -8. This means that all the "y" values on the graph for this equation are either equal to -8 or even lower. It's like a roller coaster ride that only goes down, down, down!
And there you have it! The clown has wrapped up the magic show with an explanation of the vertex, axis of symmetry, maximum or minimum value, and range of the quadratic equation y = -5(x+2)^2 - 8. Feel free to ask if you have more questions or if you'd like me to put on another magical performance!
To find the vertex, axis of symmetry, maximum or minimum value, and range of the quadratic equation y = -5(x + 2)^2 - 8, you can follow these steps:
Step 1: Identify the values of a, b, and c in the equation y = ax^2 + bx + c. In this case, a = -5, b = 0, and c = -8.
Step 2: The vertex form of a quadratic equation is given by y = a(x-h)^2 + k, where (h, k) represents the vertex. To find the vertex, you need to determine the values of h and k.
In this equation, h = -2 (opposite sign of the constant term within the parentheses) because of the (x + 2) term. Substitute this value into the equation:
y = -5(x + 2)^2 - 8
y = -5(-2 + 2)^2 - 8
y = -5(0)^2 - 8
y = 0 - 8
y = -8
So, the vertex is (-2, -8).
Step 3: To find the axis of symmetry, use the x-coordinate of the vertex. In this case, the x-coordinate is -2. Therefore, the axis of symmetry is x = -2.
Step 4: To determine whether the parabola opens upwards or downwards, you can consider the coefficient of the x^2 term, which is -5 in this equation. Since -5 is negative, the parabola opens downwards, and the vertex corresponds to the maximum value.
Step 5: The maximum value is equal to the y-coordinate of the vertex, which is -8 in this case. Therefore, the maximum value is -8.
Step 6: The domain is the set of all real numbers since there are no restrictions on the x-values.
Step 7: The range represents all possible y-values of the quadratic function. Since the parabola opens downwards and the vertex is the maximum point, the range consists of all real numbers less than or equal to the y-coordinate of the vertex, which is -8.
In summary:
- The vertex is (-2, -8)
- The axis of symmetry is x = -2
- The maximum value is -8
- The domain is all real numbers
- The range is all real numbers less than or equal to -8.
To find the vertex, axis of symmetry, maximum or minimum value, and range of a quadratic equation, we can use a process called completing the square. Here's how you can solve it step by step:
Step 1: Start with the equation y = -5(x + 2)^2 - 8.
Step 2: Identify the values of a, b, and c in the general form of a quadratic equation, y = ax^2 + bx + c. In this case, a = -5, b = 0, and c = -8.
Step 3: The vertex form of a quadratic equation is y = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. We need to convert our equation to this form.
Step 4: Expand the equation y = -5(x + 2)^2 - 8.
y = -5(x^2 + 4x + 4) - 8
y = -5x^2 - 20x - 20 - 8
y = -5x^2 - 20x - 28
Step 5: Now, we want to complete the square. To do this, focus on the terms involving x (i.e., -20x). Divide the coefficient of x by two and square it. In this case, (-20/2)^2 = 100.
Step 6: Add and subtract the value obtained from step 5 inside the parentheses. Since we divided the coefficient of x by two, we have to multiply the constant term by the same value. So, we add +100 to the equation.
y = -5x^2 - 20x + 100 - 100 - 28
Step 7: Group the squares together and simplify.
y = -5(x^2 + 4x + 100) - 128
Step 8: The equation inside the parentheses, x^2 + 4x + 100, can be rewritten as (x + 2)^2.
Step 9: Replace the equation inside the parentheses with (x + 2)^2.
y = -5(x + 2)^2 - 128
Step 10: Finally, we can identify the vertex, axis of symmetry, maximum or minimum value, and range based on the equation in vertex form.
- The vertex is given by the coordinates (-h, k) from the equation, which in this case is (-2, -128).
- The axis of symmetry is the vertical line x = -h, so the axis of symmetry is x = -2.
- The coefficient "a" in the equation (-5 in our example) determines whether the parabola opens upwards or downwards. Since a is negative, the parabola opens downwards, so the vertex represents the maximum value.
- The maximum or minimum value is given by the value of "k" in the vertex form, which is -128 in this case.
- The domain is all real numbers, as there are no restrictions on x.
- The range is all real numbers less than or equal to the y-coordinate of the vertex, which means the range is all real numbers less than or equal to -128.
So, the vertex is (-2, -128), the axis of symmetry is x = -2, the maximum value is -128, and the range is all real numbers less than or equal to -128.
y = a(x-h)^2 +k
y = -5(x+2)^2 - 8
Since a is negative, we know that the parabola will face down, so we will have a maximum and not a minimum
The vertex is given as (-h,k)
You have a -2 because it has to be the opposite of 2 and -8 for k because it is supposed to be the same sign. (-2,-8)
axis of symmetry is always = to -h which is the x-value. If you folded the parabola along that line, it would match showing its symmetry.
Since we determined earlier that the parabola is facing down, we have a maximum which is the y value of -8.
For a parabola the domain is always all real numbers. This is true because there is no possible number that will make the result undefined.
Since -8 is the greatest value for y then the range is all real numbers less than or equal to -8
You can also look at the parabola in ax^2+bx + c form. You would do that by squaring x+2 and multiplying by -5