Convert Quadratic Graph to Algebraic Equation: Finding Function with Point (5,0)

Q: How do I know if it's addition or subtraction in the equation?

Look at the vertex position! If the lowest point moved up from the origin, add the number. If it moved down, subtract. Here, vertex moved up 5 units, so it's $ +5 $.

Q: Why isn't it y = 5x² if there's a 5 in the graph?

The number 5 shows the y-coordinate where the vertex moved, not a coefficient! $ y = 5x^2 $ would make the parabola narrower, but this graph shows the same width moved up.

Q: Does the order matter in y = x² + 5?

No! You can write it as $ y = x^2 + 5 $ or $ y = 5 + x^2 $ - both are correct. The important part is identifying the vertical shift correctly.

Vertical Translation with Y-intercept Identification

Find the corresponding algebraic representation for the function

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the appropriate algebraic representation for the function

00:03 Find the intersection point of the function with the Y-axis

00:07 This is the point

00:12 Substitute X=0 in each function and compare the intersection points

00:21 According to this representation, the intersection point is different, so this is not the representation

00:27 Continue with this method for each representation and check the intersection points

00:36 According to this representation, the intersection point is different, so this is not the representation

00:45 According to this representation, the intersection point is equal, so this is the correct representation

01:00 According to this representation, the intersection point is different, so this is not the representation

01:07 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Find the corresponding algebraic representation for the function

Step-by-step solution

To solve this problem, we need to identify how the given parabola is translated from its standard form.

The standard form of a parabola is $y = x^2$ . When a vertical translation occurs, the equation becomes $y = x^2 + c$ , where $c$ shifts the parabola up or down along the y-axis.

In the graph, there is an indication that the minimum point (or vertex) of the parabola has been shifted upwards so that it crosses the $y$ -axis at the point where $y = 5$ . This tells us that the entire parabola has been shifted vertically upwards by 5 units. Therefore, $c = 5$ .

Thus, the algebraic representation of the translated function is:

$y = x^2 + 5$

This matches exactly with choice 3 from the provided options.

Therefore, the solution to the problem is $y = x^2 + 5$ .

Final Answer

$y=x^2+5$

Key Points to Remember

Essential concepts to master this topic

Translation Rule: Adding constant moves parabola up by that amount
Technique: Vertex at (0,5) means $y = x^2 + 5$
Check: At x=0, y should equal 5: $0^2 + 5 = 5$ ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal and vertical shifts
Don't think the point (5,0) means $y = 5x^2$ = wrong transformation! This confuses coefficient changes with translations. Always identify where the vertex moved: up 5 units means $y = x^2 + 5$ .

Practice Quiz

Test your knowledge with interactive questions

One function

$$ y=-6x^2 $$

to the corresponding graph:

FAQ

Everything you need to know about this question

How do I know if it's addition or subtraction in the equation?

Look at the vertex position! If the lowest point moved up from the origin, add the number. If it moved down, subtract. Here, vertex moved up 5 units, so it's $+5$ .

Why isn't it y = 5x² if there's a 5 in the graph?

The number 5 shows the y-coordinate where the vertex moved, not a coefficient! $y = 5x^2$ would make the parabola narrower, but this graph shows the same width moved up.

What if the parabola opened downward instead?

The same rule applies! A downward parabola shifted up 5 units would be $y = -x^2 + 5$ . The direction doesn't change the translation.

How can I verify my answer is correct?

Pick any point you can see clearly on the graph and substitute into your equation. For example, at $x = 0$ , the graph shows $y = 5$ , and $0^2 + 5 = 5$ ✓

Does the order matter in y = x² + 5?

No! You can write it as $y = x^2 + 5$ or $y = 5 + x^2$ - both are correct. The important part is identifying the vertical shift correctly.

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