Calculate Points A and B for the Quadratic Graph of f(x) = x² - 3x - 4

Q: How do I know which factors of -4 to choose?

Look for two numbers that multiply to -4 and add to -3. Try all factor pairs: (1,-4), (-1,4), (2,-2). Only (-4) and (1) work because (-4) × (1) = -4 and (-4) + (1) = -3.

Q: Why do we set the quadratic equal to zero?

The x-intercepts are where the graph crosses the x-axis, meaning the y-value (or f(x)) equals zero at those points. Setting $ f(x) = 0 $ finds these crossing points.

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

You can always use the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $. For this problem: a=1, b=-3, c=-4, which gives the same answers: x = -1 and x = 4.

Q: How do I tell which point is A and which is B from the graph?

Look at the position on the x-axis. Point A is to the left of the y-axis at x = -1, and point B is to the right at x = 4. The coordinates are A(-1,0) and B(4,0).

Q: Do the y-coordinates of A and B always equal zero?

Yes! Points A and B are x-intercepts, which means they lie on the x-axis where y = 0. All x-intercepts have coordinates in the form (x, 0).

Q: Can I check my factoring without substituting back?

Yes! Expand your factored form: $ (x+1)(x-4) = x^2 - 4x + x - 4 = x^2 - 3x - 4 $. If it matches the original equation, your factoring is correct.

Quadratic Factoring with X-Intercept Identification

The following function has been graphed below:

$f(x)=x^2-3x-4$

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 intersection points with X-axis

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

00:12 Substitute Y = 0 and solve for X values

00:19 Break down the function into a trinomial

00:24 This is the corresponding trinomial

00:30 Find what zeroes each factor in the product

00:34 This is one solution

00:37 This is the second solution

00:46 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:

$f(x)=x^2-3x-4$

Calculate points A and B.

Step-by-step solution

To solve for points A and B, we find the x-intercepts of the function by setting:

$f(x) = x^2 - 3x - 4 = 0$

We check if it can be factored:

Factor $x^2 - 3x - 4$ . The factors of -4 that add to -3 are -4 and 1.

Thus, factor the function as $(x - 4)(x + 1) = 0$ .

Set each factor to zero:

$x - 4 = 0$ implies $x = 4$
$x + 1 = 0$ implies $x = -1$

These are the x-intercepts, or roots, of the quadratic function.

Therefore, the coordinates of points A and B, where the function intersects the x-axis, are $A(-1, 0)$ and $B(4, 0)$ .

The correct choice corresponding to these points is option 3: $A(-1,0), B(4,0)$ .

Thus, the solution to the problem is $A(-1,0), B(4,0)$ .

Final Answer

$A(-1,0),B(4,0)$

Key Points to Remember

Essential concepts to master this topic

X-Intercepts: Set quadratic equal to zero and solve for x-values
Factoring: Find factors of -4 that sum to -3: (-4) + (1) = -3
Verification: Substitute x = -1 and x = 4 into original equation: both give f(x) = 0 ✓

Common Mistakes

Avoid these frequent errors

Incorrectly identifying the factors that add to the middle coefficient
Don't find factors of -4 like (-2) and (2) that don't add to -3 = wrong factorization! These factors multiply to give -4 but add to 0, not -3. Always check that your factors both multiply to the constant term AND add to the middle coefficient.

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

How do I know which factors of -4 to choose?

Look for two numbers that multiply to -4 and add to -3. Try all factor pairs: (1,-4), (-1,4), (2,-2). Only (-4) and (1) work because (-4) × (1) = -4 and (-4) + (1) = -3.

Why do we set the quadratic equal to zero?

The x-intercepts are where the graph crosses the x-axis, meaning the y-value (or f(x)) equals zero at those points. Setting $f(x) = 0$ finds these crossing points.

What if I can't factor the quadratic easily?

You can always use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . For this problem: a=1, b=-3, c=-4, which gives the same answers: x = -1 and x = 4.

How do I tell which point is A and which is B from the graph?

Look at the position on the x-axis. Point A is to the left of the y-axis at x = -1, and point B is to the right at x = 4. The coordinates are A(-1,0) and B(4,0).

Do the y-coordinates of A and B always equal zero?

Yes! Points A and B are x-intercepts, which means they lie on the x-axis where y = 0. All x-intercepts have coordinates in the form (x, 0).

Can I check my factoring without substituting back?

Yes! Expand your factored form: $(x+1)(x-4) = x^2 - 4x + x - 4 = x^2 - 3x - 4$ . If it matches the original equation, your factoring is correct.

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