Quadratic Equation: Finding Algebraic Expression for Point (5,-4)

Q: Why is it (x - 5) and not (x + 5) when the x-coordinate is 5?

In vertex form $ y = (x - h)^2 + k $, the h value gets subtracted from x. So when the vertex x-coordinate is 5, we write (x - 5). Think of it as: what makes the expression zero? When x = 5, then (x - 5) = 0.

Q: How do I know if the parabola opens up or down?

Look at the coefficient in front of the squared term. If it's positive (like +1 in our case), the parabola opens upward. If it's negative, it opens downward. Our equation $ y = (x-5)^2 - 4 $ opens up!

Q: What if I can't see the vertex clearly in the graph?

Look for the lowest point (if parabola opens up) or highest point (if opens down). The coordinates given as (5,-4) tell you this is the vertex. You can also find where the parabola changes direction.

Q: Can I expand this form to standard form?

Yes! Expand $ y = (x-5)^2 - 4 $ to get $ y = x^2 - 10x + 25 - 4 = x^2 - 10x + 21 $. But vertex form is often more useful for identifying the vertex directly.

Vertex Form with Coordinate Identification

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

00:05 The function is a parabola

00:13 The minimum point is 5 units to the right according to the drawing, thus we'll find P

01:01 The minimum point is 4 units down according to the drawing, thus we'll find K

01:27 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 solve this problem, we'll use the vertex form of a parabola equation, $y = (x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Step 1: We have the point $(5, -4)$ that indicates the vertex of the parabola.

Step 2: Substitute $h = 5$ and $k = -4$ into the vertex form equation.

By substitution, the equation becomes:

$y = (x - 5)^2 - 4$

Therefore, the algebraic representation of the parabola is $\mathbf{y = (x - 5)^2 - 4}$ .

Final Answer

$y=(x-5)^2-4$

Key Points to Remember

Essential concepts to master this topic

Vertex Form: Use $y = (x - h)^2 + k$ where (h,k) is the vertex
Substitution: Replace h = 5 and k = -4 to get $y = (x - 5)^2 - 4$
Verification: Check that point (5,-4) satisfies equation: $-4 = (5-5)^2 - 4 = -4$ ✓

Common Mistakes

Avoid these frequent errors

Confusing signs in vertex form substitution
Don't write y = (x + 5)² + 4 when vertex is (5,-4) = wrong parabola position! This creates a vertex at (-5,4) instead of (5,-4). Always use y = (x - h)² + k and substitute h = 5, k = -4 directly.

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 is it (x - 5) and not (x + 5) when the x-coordinate is 5?

In vertex form $y = (x - h)^2 + k$ , the h value gets subtracted from x. So when the vertex x-coordinate is 5, we write (x - 5). Think of it as: what makes the expression zero? When x = 5, then (x - 5) = 0.

How do I know if the parabola opens up or down?

Look at the coefficient in front of the squared term. If it's positive (like +1 in our case), the parabola opens upward. If it's negative, it opens downward. Our equation $y = (x-5)^2 - 4$ opens up!

What if I can't see the vertex clearly in the graph?

Look for the lowest point (if parabola opens up) or highest point (if opens down). The coordinates given as (5,-4) tell you this is the vertex. You can also find where the parabola changes direction.

Can I expand this form to standard form?

Yes! Expand $y = (x-5)^2 - 4$ to get $y = x^2 - 10x + 25 - 4 = x^2 - 10x + 21$ . But vertex form is often more useful for identifying the vertex directly.

How do I check my answer is correct?

Substitute the vertex coordinates into your equation. For (5,-4): $y = (5-5)^2 - 4 = 0 - 4 = -4$ ✓. The y-value should match the vertex y-coordinate!

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