
Part 2 of 2

Question content area left

Part 1

Micah found the vertex for the function 

y equals negative 9.5 x squared minus 47.5 x plus 63y=−9.5x2−47.5x+63 

as shown. Find and correct​ Micah's error.

xequals=negative StartFraction b Over 2 a EndFraction−b2a

xequals=negative StartFraction 47.5 Over 2 left parenthesis negative 9.5 right parenthesis EndFraction−47.52(−9.5)

xequals=negative StartFraction 47.5 Over negative 19 EndFraction−47.5−19

xequals=negative left parenthesis negative 2.5 right parenthesis−(−2.5)

xequals=2.5

yequals=negative 9.5 left parenthesis 2.5 right parenthesis squared minus 47.5 left parenthesis 2.5 right parenthesis plus 63−9.5(2.5)2−47.5(2.5)+63

yequals=negative 59.375 minus 118.75 plus 63−59.375−118.75+63

yequals=minus−115.125

...

Question content area right

Part 1

Explain the error.

A.

Micah should have evaluated the function with 

xequals=0 

to find the​ y-coordinate.

B.

Micah did not use the correct​ order-of-operations dividing 47.5 by 

2 left parenthesis negative 9.5 right parenthesis2(−9.5). 

C.

Micah used the wrong sign for b in the formula 

xequals=negative StartFraction b Over 2 a EndFraction−b2a.

Your answer is correct.

D.

Micah should have found a positive value when he simplified the 

negative 9.5 left parenthesis 2.5 right parenthesis squared−9.5(2.5)2 

term. 

Part 2

The correct vertex is 

enter your response here.

​(Type an ordered​ pair.)

Explain the error.The correct vertex is(Type an ordered pair.)

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Let's analyze Micah's work step by step to identify the error and find the correct vertex for the function:

### Given function:
\[
y = -9.5x^2 - 47.5x + 63
\]

### Step 1: Finding the x-coordinate of the vertex
The formula for finding the x-coordinate of the vertex for a quadratic function \( y = ax^2 + bx + c \) is:

\[
x = -\frac{b}{2a}
\]

For the given function, \( a = -9.5 \) and \( b = -47.5 \).

Micah's calculation:
\[
x = -\frac{47.5}{2(-9.5)}
\]
\[
x = -\frac{47.5}{-19}
\]
\[
x = -(-2.5)
\]
\[
x = 2.5
\]

**Error:**
Micah incorrectly used \( b = 47.5 \) instead of \( b = -47.5 \) in the formula. The correct substitution should be:

\[
x = -\frac{-47.5}{2(-9.5)}
\]
\[
x = \frac{47.5}{-19}
\]
\[
x = -2.5
\]

So, the correct x-coordinate of the vertex is \( x = -2.5 \), not \( x = 2.5 \).

### Step 2: Finding the y-coordinate of the vertex

Substitute \( x = -2.5 \) into the original equation to find the y-coordinate:

\[
y = -9.5(-2.5)^2 - 47.5(-2.5) + 63
\]

Calculate each term:
\[
y = -9.5(6.25) + 118.75 + 63
\]
\[
y = -59.375 + 118.75 + 63
\]
\[
y = 122.375 - 59.375
\]
\[
y = 63
\]

So, the y-coordinate of the vertex is \( y = 63 \).

### Conclusion:
The correct vertex is \( (-2.5, 63) \).

### Explanation of the Error:
**Correct answer: C. Micah used the wrong sign for \( b \) in the formula \( x = -\frac{b}{2a} \).**

**Correct vertex:** The correct vertex is \((-2.5, 63)\).

Would you like more details, or do you have any other questions?

### Follow-up Questions:
1. How does the vertex form of a quadratic equation relate to the standard form?
2. What happens to the vertex when the coefficient \(a\) is positive instead of negative?
3. How would you find the vertex of a quadratic function if the equation were given in factored form?
4. What is the significance of the vertex in the graph of a parabola?
5. How does changing the value of \(c\) in the quadratic equation affect the vertex?

**Tip:** Always double-check the signs when substituting values into formulas, especially in quadratic equations, where sign errors can lead to incorrect vertex calculations.

Explore how to find and correct errors in calculating the vertex of a quadratic function, focusing on Micah's mistake in using the wrong sign for 'b'. Includes step-by-step solution and explanation.

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