Write the quadratic function (or one possible quadratic function ) in vertex form or standard form, from the following information:
zeros at (0,0) and (7,0), and opening down
I know the answer is
-x(x-7)=-x^2+7x
but I don't get it
please baby talk
thank you and sorry
the zeroes mean that
y = a(x)(x-7)
since it opens downward, a must be negative
so, a = -1 is the simplest choice
Oobleck, but how do I know how to write it,
Vertex form is a(x-h)^2+k
and standard form is:
ax^2+bx+c
so how does the zeros at (0,0) and (7,0), and opening down
end up like this as the answer:
-x(x-7)=-x^2+7x
sorry, I am really bad at this
No problem! Let's break it down step by step in baby talk:
1. First, we're given that the quadratic function has zeros at (0,0) and (7,0). "Zeros" mean the x-values where the quadratic function equals zero, in other words, where it crosses the x-axis.
So, we know that the quadratic function crosses the x-axis at x = 0 and x = 7. This means those are the two points where the quadratic function touches or intersects the x-axis.
2. Next, we're told that the quadratic function opens downward. This means the parabola formed by the function curves downwards.
3. We want to write the function in either vertex form (also known as the completed square form) or standard form. These are two different ways to express a quadratic function.
4. Let's start by using standard form. In standard form, the quadratic function looks like this: ax^2 + bx + c = 0. We want to find the values of a, b, and c.
5. We're given that the two zeros (x-intercepts) are at (0,0) and (7,0). This means those are the two x-values where the quadratic function equals zero.
6. Using these zeros, we can write two equations:
- When x = 0, the equation becomes 0a + 0b + c = 0 (since x = 0, the x^2 and x terms become zero). Simplifying, we get c = 0.
- When x = 7, the equation becomes 49a + 7b + c = 0 (since x = 7, we substitute it in). We already know c = 0, so this simplifies to 49a + 7b = 0.
7. Now, let's use the fact that the quadratic function opens downward. This means the coefficient of the x^2 term (a) should be negative.
To achieve this, we assign a value of -1 to a.
8. Substituting these values into the equation we got in step 6, we have:
49(-1) + 7b = 0 (since a = -1). Simplifying further, we get -49 + 7b = 0.
9. Solving this equation, we find that b = 7.
10. Now we have the values of a, b, and c. Plugging them into the standard form equation, we get:
-x^2 + 7x + 0 = 0 (since c = 0).
11. Finally, we simplify the equation by removing the unnecessary zeros:
-x^2 + 7x = 0.
So, the quadratic function with zeros at (0,0) and (7,0), and opening downward, can be written as -x^2 + 7x.