Solve for A and B: Analyzing the Quadratic Graph of f(x) = x² - 6x

Q: What if I can't factor the quadratic?

If factoring doesn't work easily, you can always use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. But try factoring first - it's usually faster!

Q: How do I know which point is A and which is B?

Look at the graph! Point A appears to be at the origin (leftmost intersection), so A = $ (0,0) $. Point B is further right, so B = $ (6,0) $.

Q: Why does this parabola pass through the origin?

Because there's no constant term in $ f(x) = x^2 - 6x $! When $ x = 0 $, we get $ f(0) = 0^2 - 6(0) = 0 $, so the graph passes through $ (0,0) $.

Finding x-intercepts with Factoring Methods

The following function has been graphed below:

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

Calculate points A and 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 points A,B

00:03 Notice that points A,B are the intersection points with the X-axis

00:07 At the intersection points with X-axis, Y value must = 0

00:14 Substitute Y = 0 and solve for X values

00:19 Factor out the common term from the parentheses

00:22 Find what makes each factor equal zero

00:25 This is one solution

00:29 This is second solution

00:32 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:

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

Calculate points A and B.

Step-by-step solution

To solve for the points A and B, where the graph of $f(x) = x^2 - 6x$ intersects the x-axis, let's solve the equation $f(x) = 0$ :
1. Write the equation in standard form:
$x^2 - 6x = 0$ .

2. Factor the quadratic equation:
$x(x - 6) = 0$ .

3. Set each factor equal to zero:
$x = 0$ or $x - 6 = 0$ .

4. Solve each equation for $x$ :
$x = 0$ and $x = 6$ .

Thus, the points where the function intersects the x-axis, also the points A and B, are $(0,0)$ and $(6,0)$ .

Therefore, the solution to the problem is $(6,0), (0,0)$ .

Final Answer

$(6,0),\lparen0,0)$

Key Points to Remember

Essential concepts to master this topic

X-intercepts: Set function equal to zero and solve for x
Factoring: Factor out common x: $x^2 - 6x = x(x - 6) = 0$
Check: Substitute back: $0^2 - 6(0) = 0$ and $6^2 - 6(6) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to set the function equal to zero
Don't try to find where the graph crosses the x-axis without setting $f(x) = 0$ = you'll get the wrong points! The x-intercepts occur only where the function equals zero. Always start by writing $x^2 - 6x = 0$ before factoring.

Practice Quiz

Test your knowledge with interactive questions

The following function has been graphed below:

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

Calculate points A and B.

FAQ

Everything you need to know about this question

Why do I need to factor out the x first?

Factoring out x makes the equation easier to solve! When you have $x(x - 6) = 0$ , you can use the zero product property: if two things multiply to zero, at least one must be zero.

What if I can't factor the quadratic?

If factoring doesn't work easily, you can always use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . But try factoring first - it's usually faster!

Are points A and B always on the x-axis?

Yes! When a problem asks for points where a graph intersects the x-axis, the y-coordinate is always 0. That's why we get $(0,0)$ and $(6,0)$ .

How do I know which point is A and which is B?

Look at the graph! Point A appears to be at the origin (leftmost intersection), so A = $(0,0)$ . Point B is further right, so B = $(6,0)$ .

Why does this parabola pass through the origin?

Because there's no constant term in $f(x) = x^2 - 6x$ ! When $x = 0$ , we get $f(0) = 0^2 - 6(0) = 0$ , so the graph passes through $(0,0)$ .

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