Locate Point C on f(x) = -x² + 5x + 6: Quadratic Graph Analysis

Q: Why is the vertex formula x = -b/(2a)?

The vertex formula comes from completing the square or using calculus. For any parabola $ ax^2 + bx + c $, the vertex always occurs at $ x = -\frac{b}{2a} $. It's a reliable shortcut!

Q: How do I remember which coefficient is which?

In standard form $ ax^2 + bx + c $: a is with x², b is with x, and c is the constant. For $ f(x) = -x^2 + 5x + 6 $, we have a = -1, b = 5, c = 6.

Q: What if I get a decimal for the x-coordinate?

Decimals are perfectly normal for vertex coordinates! $ x = 2.5 $ just means the vertex is halfway between x = 2 and x = 3. Always convert to mixed numbers if the answer choices show them that way.

Q: Do I always need to find the y-coordinate too?

Yes! The vertex is a point, so you need both coordinates. After finding x = 2.5, substitute it back: $ f(2.5) = -(2.5)^2 + 5(2.5) + 6 = 12.25 $.

Q: Why is point C at the top of the parabola?

Since the coefficient of x² is negative (a = -1), this parabola opens downward. The vertex is the highest point, which makes sense for point C's position in the graph.

Vertex Formula with Quadratic Coordinates

The following function has been graphed below:

$f(x)=-x^2+5x+6$

Calculate point C.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the coordinates of point C

00:03 Point C is the vertex point so we want to find it

00:07 We'll use the formula to calculate the vertex point

00:10 Let's look at the function coefficients

00:16 We'll substitute appropriate values and solve for X

00:26 This is the X value at point C

00:32 Now we'll substitute this value in the function to find Y value at point C

00:37 Let's calculate and solve

00:52 This is the Y value at point C

00:56 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The following function has been graphed below:

$f(x)=-x^2+5x+6$

Calculate point C.

Step-by-step solution

To answer the question, we must first remember the formula for finding the vertex of a parabola:

Now let's substitute the known data into the formula:

-5/2(-1)=-5/-2=2.5

In other words, the x-coordinate of the vertex of the parabola is found when the X value equals 2.5.

Now let's substitute this into the parabola equation to find the Y value:

-(2.5)²+5*2.5+6= 12.25

Therefore, the coordinates of the vertex of the parabola are (2.5, 12.25).

Final Answer

$(2\frac{1}{2},12\frac{1}{4})$

Key Points to Remember

Essential concepts to master this topic

Vertex Formula: For ax² + bx + c, x-coordinate is -b/(2a)
Technique: Substitute x = -5/(2(-1)) = 2.5 into original function
Check: f(2.5) = -(2.5)² + 5(2.5) + 6 = 12.25 ✓

Common Mistakes

Avoid these frequent errors

Using wrong sign in vertex formula
Don't forget the negative sign: x = b/(2a) = wrong answer! The vertex formula is x = -b/(2a), not +b/(2a). This sign error moves the vertex to the wrong side of the y-axis. Always use x = -b/(2a) with the negative sign.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

$$ f(x)=x^2-6x $$

Calculate point C.

FAQ

Everything you need to know about this question

Why is the vertex formula x = -b/(2a)?

The vertex formula comes from completing the square or using calculus. For any parabola $ax^2 + bx + c$ , the vertex always occurs at $x = -\frac{b}{2a}$ . It's a reliable shortcut!

How do I remember which coefficient is which?

In standard form $ax^2 + bx + c$ : a is with x², b is with x, and c is the constant. For $f(x) = -x^2 + 5x + 6$ , we have a = -1, b = 5, c = 6.

What if I get a decimal for the x-coordinate?

Decimals are perfectly normal for vertex coordinates! $x = 2.5$ just means the vertex is halfway between x = 2 and x = 3. Always convert to mixed numbers if the answer choices show them that way.

Do I always need to find the y-coordinate too?

Yes! The vertex is a point, so you need both coordinates. After finding x = 2.5, substitute it back: $f(2.5) = -(2.5)^2 + 5(2.5) + 6 = 12.25$ .

Why is point C at the top of the parabola?

Since the coefficient of x² is negative (a = -1), this parabola opens downward. The vertex is the highest point, which makes sense for point C's position in the graph.

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