Calculate Point B on the Graph of f(x) = x² - 6x + 8

Q: Why is the vertex important for quadratic functions?

The vertex is the turning point of the parabola - either the highest or lowest point on the graph. It tells you the maximum or minimum value of the function.

Q: How do I know if the vertex is a maximum or minimum?

Look at the coefficient of $ x^2 $! If it's positive (like +1 in our example), the parabola opens upward and the vertex is a minimum. If negative, it opens downward and the vertex is a maximum.

Q: What if I can't remember the vertex formula?

You can also complete the square or find where the derivative equals zero. But memorizing $ x = -\frac{b}{2a} $ is much faster for tests!

Q: What does 'a = 1' and 'b = -6' mean?

These come from the standard form $ ax^2 + bx + c $. In $ f(x) = x^2 - 6x + 8 $: a = 1 (coefficient of $ x^2 $), b = -6 (coefficient of x), and c = 8 (constant term).

Quadratic Functions with Vertex Identification

The following function has been graphed below.

$f(x)=x^2-6x+8$

Calculate point B.

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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 B

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

00:08 We'll use the formula to calculate the vertex point

00:11 Let's identify the function coefficients

00:17 We'll substitute appropriate values and solve for X

00:25 This is the X value at point C

00:30 Now we'll substitute this value in the function to find the Y value at point C

00:35 Let's calculate and solve

00:41 This is the Y value at point C

00:45 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.

$f(x)=x^2-6x+8$

Calculate point B.

Step-by-step solution

To calculate point B, we should determine the vertex of the quadratic function $f(x) = x^2 - 6x + 8$ .

The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$ .

In our equation, we have $a = 1$ and $b = -6$ , therefore:

$x = -\frac{-6}{2 \times 1} = \frac{6}{2} = 3$

Next, we substitute $x = 3$ back into the function to find the y-coordinate:

$f(3) = 3^2 - 6 \times 3 + 8 = 9 - 18 + 8 = -1$

Thus, the vertex, which is point B, is $(3, -1)$ .

Therefore, the solution indicates that point B is at $(3, -1)$ .

Final Answer

$(3,-1)$

Key Points to Remember

Essential concepts to master this topic

Vertex Formula: Use $x = -\frac{b}{2a}$ to find x-coordinate of vertex
Technique: For $f(x) = x^2 - 6x + 8$ , vertex x is $-\frac{-6}{2(1)} = 3$
Check: Substitute x = 3: $f(3) = 9 - 18 + 8 = -1$ , so vertex is (3, -1) ✓

Common Mistakes

Avoid these frequent errors

Using wrong signs in vertex formula
Don't forget the negative sign in $x = -\frac{b}{2a}$ = wrong x-coordinate! Students often calculate $\frac{-6}{2} = -3$ instead of $-\frac{(-6)}{2} = 3$ . Always apply the negative sign to the entire fraction $\frac{b}{2a}$ .

Practice Quiz

Test your knowledge with interactive questions

The following function has been plotted on the graph below:

$$ f(x)=x^2-8x+16 $$

Calculate point C.

FAQ

Everything you need to know about this question

Why is the vertex important for quadratic functions?

The vertex is the turning point of the parabola - either the highest or lowest point on the graph. It tells you the maximum or minimum value of the function.

How do I know if the vertex is a maximum or minimum?

Look at the coefficient of $x^2$ ! If it's positive (like +1 in our example), the parabola opens upward and the vertex is a minimum. If negative, it opens downward and the vertex is a maximum.

What if I can't remember the vertex formula?

You can also complete the square or find where the derivative equals zero. But memorizing $x = -\frac{b}{2a}$ is much faster for tests!

Can I just read the vertex from the graph?

Yes, but be careful with accuracy! The graph shows Point B at approximately (3, -1), but calculating gives you the exact coordinates. Always verify graphical readings with calculations.

What does 'a = 1' and 'b = -6' mean?

These come from the standard form $ax^2 + bx + c$ . In $f(x) = x^2 - 6x + 8$ : a = 1 (coefficient of $x^2$ ), b = -6 (coefficient of x), and c = 8 (constant term).

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