Quadratic Equation: Finding Algebraic Form for Arc Through (-10,-3)

Q: Why does the vertex (-10,-3) become (x+10) instead of (x-10)?

In vertex form $ y = (x - h)^2 + k $, we substitute h = -10. So we get $ (x - (-10))^2 $, which simplifies to $ (x + 10)^2 $. The double negative creates a plus sign!

Q: How do I remember which coordinate goes where?

Think "h for horizontal, k for up"! The first coordinate (x-value) goes in the h position with (x - h), and the second coordinate (y-value) goes in the k position as + k.

Q: What if I accidentally use the wrong signs?

Your parabola will be in the wrong location! Always check by substituting the vertex coordinates back into your equation. If you get the y-coordinate of the vertex, you're correct.

Q: Can I expand (x+10)² to check my answer?

Yes! $ (x+10)^2 - 3 = x^2 + 20x + 100 - 3 = x^2 + 20x + 97 $. But vertex form is usually more useful because it shows the vertex clearly.

Q: Does the parabola open up or down?

This parabola opens upward because there's no negative sign in front of the squared term. The coefficient of $ (x+10)^2 $ is positive 1.

Vertex Form with Negative Coordinates

Find the corresponding algebraic representation of the drawing:

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Step-by-step video solution

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00:00 Find the appropriate algebraic representation for the function

00:03 The function is a parabola

00:23 The minimum point is 10 units left according to the drawing, that's how we find P

01:17 The minimum point is 3 units down according to the drawing, that's how we find K

01:36 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

Find the corresponding algebraic representation of the drawing:

Step-by-step solution

To determine the algebraic representation of the parabola, follow these steps:

Step 1: Recognize the parabola vertex form, $y = (x - p)^2 + k$ , which uses $(p, k)$ as the vertex.
Step 2: Given that the vertex (from the drawing) is $(-10, -3)$ , substitute these values into $p$ and $k$ .
Step 3: Utilizing vertex form, substitute $p = -10$ and $k = -3$ into the expression, resulting in: $y = (x + 10)^2 - 3$ .

As a result, the parabola is represented algebraically by replacing $(x-p)^2$ with $(x - (-10))^2$ , simplifying to $(x + 10)^2$ , and adding $k$ , i.e., $-3$ .

Therefore, the equation that corresponds with the drawing is $y = (x + 10)^2 - 3$ .

Final Answer

$y=(x+10)^2-3$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Use $y = (x - h)^2 + k$ where vertex is (h,k)
Sign Substitution: For vertex (-10,-3), get $y = (x - (-10))^2 + (-3)$
Verification: Check vertex point: when x = -10, y should equal -3 ✓

Common Mistakes

Avoid these frequent errors

Confusing signs when substituting negative coordinates
Don't write $y = (x - 10)^2 - 3$ for vertex (-10,-3) = wrong parabola shifted right instead of left! The negative coordinate becomes positive in the expression. Always substitute carefully: $(x - (-10)) = (x + 10)$ .

Practice Quiz

Test your knowledge with interactive questions

Which equation represents the function:

$$ y=x^2 $$

moved 2 spaces to the right

and 5 spaces upwards.

FAQ

Everything you need to know about this question

Why does the vertex (-10,-3) become (x+10) instead of (x-10)?

In vertex form $y = (x - h)^2 + k$ , we substitute h = -10. So we get $(x - (-10))^2$ , which simplifies to $(x + 10)^2$ . The double negative creates a plus sign!

How do I remember which coordinate goes where?

Think "h for horizontal, k for up"! The first coordinate (x-value) goes in the h position with (x - h), and the second coordinate (y-value) goes in the k position as + k.

What if I accidentally use the wrong signs?

Your parabola will be in the wrong location! Always check by substituting the vertex coordinates back into your equation. If you get the y-coordinate of the vertex, you're correct.

Can I expand (x+10)² to check my answer?

Yes! $(x+10)^2 - 3 = x^2 + 20x + 100 - 3 = x^2 + 20x + 97$ . But vertex form is usually more useful because it shows the vertex clearly.

Does the parabola open up or down?

This parabola opens upward because there's no negative sign in front of the squared term. The coefficient of $(x+10)^2$ is positive 1.

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