Find the Quadratic Equation: Converting Graph with Point (-2,7)

Q: Why does the vertex form use (x - h) when my vertex has a negative x-coordinate?

The vertex form is $ y = (x - h)^2 + k $. When h = -2, you get $ y = (x - (-2))^2 + k $ which simplifies to $ y = (x + 2)^2 + k $. Always subtract h, even if h is negative!

Q: What if I'm given a point that's not the vertex?

You need additional information like the vertex location or another point. With just one non-vertex point, you can't determine the unique equation since many parabolas can pass through a single point.

Q: Can I expand the vertex form to standard form?

Yes! $ y = (x + 2)^2 + 7 $ expands to $ y = x^2 + 4x + 4 + 7 = x^2 + 4x + 11 $. Both forms represent the same parabola.

Vertex Form Equations with Given Point

Find the corresponding algebraic representation of the drawing:

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Step-by-step video solution

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00:00 Find appropriate algebraic representation for the function

00:04 The function is a parabola

00:22 The minimum point is 2 units to the left according to the drawing, that's how we find P

01:02 The minimum point is 7 units up according to the drawing, that's how we find K

01:23 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, we use the vertex form of a parabola, which is $y = (x-h)^2 + k$ . Here, the vertex is placed at $(-2, 7)$ , thus plug these values into our equation: $h = -2$ and $k = 7$ .

Consequently, the equation of the parabola becomes:

$y = (x + 2)^2 + 7$

This representation correctly describes a parabola that passes through the vertex at $(-2, 7)$ and opens upwards, as indicated by the absence of a negative sign or alternate coefficient in front of the square term.

Therefore, the correct choice corresponding to this problem formulation is:

$y = (x + 2)^2 + 7$

Final Answer

$y=(x+2)^2+7$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Use y = (x - h)² + k where (h,k) is vertex
Technique: For vertex (-2,7), substitute h = -2 and k = 7
Check: Verify point (-2,7) satisfies equation: y = (-2+2)² + 7 = 7 ✓

Common Mistakes

Avoid these frequent errors

Confusing signs in vertex form substitution
Don't write y = (x - (-2))² + 7 as y = (x - 2)² + 7! This gives wrong vertex (2,7) instead of (-2,7). Always remember y = (x - h)² + k means subtract h, so for h = -2, write y = (x - (-2))² = (x + 2)².

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 form use (x - h) when my vertex has a negative x-coordinate?

The vertex form is $y = (x - h)^2 + k$ . When h = -2, you get $y = (x - (-2))^2 + k$ which simplifies to $y = (x + 2)^2 + k$ . Always subtract h, even if h is negative!

How can I tell if the parabola opens up or down from the graph?

Look at the shape! If the parabola looks like a U (opens upward), the coefficient of the squared term is positive. If it looks like an upside-down U (opens downward), the coefficient is negative.

What if I'm given a point that's not the vertex?

You need additional information like the vertex location or another point. With just one non-vertex point, you can't determine the unique equation since many parabolas can pass through a single point.

Can I expand the vertex form to standard form?

Yes! $y = (x + 2)^2 + 7$ expands to $y = x^2 + 4x + 4 + 7 = x^2 + 4x + 11$ . Both forms represent the same parabola.

How do I know which answer choice is correct?

Substitute the given point into each equation. Only the correct equation will make both sides equal when you plug in x = -2 and check if y = 7.

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