For the graph y= ax^2 + bx + c, show that the y-coordinate of the vertex is
-b^2/4a + c.
since the vertex occurs at x = -b/2a,
y = a(-b/2a)^2 + b(-b/2a) + c
= b^2/4a - b^2/2a + b
= -b^2/4a + c
thanks
To find the y-coordinate of the vertex of a quadratic function given by y = ax^2 + bx + c, we can use the formula -b^2/4a + c.
Step 1: The vertex of a quadratic function is given by the x-coordinate (-b/2a) and y-coordinate (f(-b/2a)).
Step 2: To find the x-coordinate of the vertex, we use the formula x = -b / 2a. For our function y = ax^2 + bx + c, the x-coordinate of the vertex is -b / 2a.
Step 3: Now, substitute the x-coordinate into the function to find the y-coordinate of the vertex. Replace x with -b / 2a in the function.
So, f(-b/2a) = a(-b/2a)^2 + b(-b/2a) + c.
Simplifying, we get:
f(-b/2a) = a(b^2/4a^2) - b^2/2a + c.
Step 4: Further simplifying, we can cancel out terms and combine like terms:
f(-b/2a) = b^2/4a - b^2/2a + c.
Step 5: Common denominator:
f(-b/2a) = b^2/4a - 2b^2/4a + 4ac/4a.
Step 6: Adding the numerators together:
f(-b/2a) = (b^2 - 2b^2 + 4ac)/4a.
Step 7: Simplifying the numerator:
f(-b/2a) = (-b^2 + 4ac)/4a.
Step 8: Simplifying further, we have:
f(-b/2a) = (-b^2 + 4ac)/(4a).
Step 9: Factoring out a negative sign from the numerator:
f(-b/2a) = - (b^2 - 4ac)/(4a).
Step 10: Recall that -(b^2 - 4ac) is the discriminant D, which is always non-negative for a quadratic function (since it determines the nature of the solutions).
So, we can rewrite it as:
f(-b/2a) = -D/(4a).
Step 11: Finally, since -D/(4a) is equivalent to -b^2/(4a), we have:
f(-b/2a) = -b^2/(4a).
Therefore, the y-coordinate of the vertex of the quadratic function y = ax^2 + bx + c is -b^2/(4a).
To find the y-coordinate of the vertex for the graph of a quadratic equation in the form y = ax^2 + bx + c, we can use a process known as "completing the square." Here's how to do it step by step:
Step 1: Rewrite the equation in the form y = a(x-h)^2 + k
By completing the square, we can rewrite the quadratic equation in vertex form by grouping the x terms and completing the square on the x terms. The vertex form of a quadratic equation is given by y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
Step 2: Rearrange the equation.
Rearranging the original equation, we get y = ax^2 + bx + c.
Step 3: Factor out a from the x terms.
Factor out the common factor 'a' from the x terms, which gives y = a(x^2 + (b/a)x) + c.
Step 4: Compute the value to complete the square.
To complete the square, calculate the value (b/2a)^2, which is (b^2)/(4a^2).
Step 5: Add and subtract the value from step 4 within the parentheses.
Add and subtract the value obtained in step 4 within the parentheses. This addition and subtraction will allow you to complete the square.
y = a(x^2 + (b/a)x + (b^2)/(4a^2) - (b^2)/(4a^2)) + c.
Step 6: Rewrite the equation by grouping.
Group the terms within the parentheses as a perfect square trinomial.
y = a(x^2 + (b/a)x + (b^2)/(4a^2) - (b^2)/(4a^2)) + c
= a((x + b/(2a))^2 - (b^2)/(4a^2)) + c.
Step 7: Simplify.
Simplify the expression inside the parentheses.
y = a((x + b/(2a))^2 - (b^2)/(4a^2)) + c
= a(x + b/(2a))^2 - (b^2)/(4a) + c.
Step 8: Distribute 'a' and simplify further.
Distribute 'a' to both terms inside the parentheses, and simplify the expression.
y = a(x + b/(2a))^2 - (b^2)/(4a) + c
= ax + b/2 - (b^2)/(4a) + c.
Step 9: Combine like terms.
Combine the constant terms (b/2 - (b^2)/(4a) + c) to get the simplified equation in vertex form.
y = ax + b/2 - (b^2)/(4a) + c
= ax - (b^2)/(4a) + b/2 + c.
Step 10: Rearrange the terms.
Rearrange the terms to get the vertex form.
y = -(b^2)/(4a) + b/2 + ax + c
= -(b^2)/(4a) + (2ab)/(4a) + (4ac)/(4a)
= -(b^2 + 4ac)/(4a) + (2ab)/(4a)
= (2ab - b^2 - 4ac)/(4a).
Finally, we simplify the expression to get the y-coordinate of the vertex.
y = -(b^2 + 4ac)/(4a)
= -b^2/(4a) - 4ac/(4a)
= -b^2/(4a) - c/a.
Hence, the y-coordinate of the vertex for the graph of the quadratic equation y = ax^2 + bx + c is -b^2/(4a) - c/a, which matches the expression you were looking to prove.