Convert Quadratic Graph to Algebraic Form: Finding Equation for r=128.048

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

Look at the direction of the curve! If it opens upward like a U-shape, the coefficient of $ x^2 $ is positive. If it opens downward like an upside-down U, the coefficient is negative.

Q: What does the vertex tell me about the equation?

The vertex gives you the constant term and the axis of symmetry. For vertex at (0, 1), the equation has +1 as the constant term, making it $ y = -x^2 + 1 $.

Q: Why is the coefficient of x² equal to -1 in this problem?

The graph shows a standard parabola that's flipped upside down. When there's no stretching or compressing, the coefficient is simply -1 for downward opening parabolas.

Q: How do I verify my equation is correct?

Test key points! Check that when $ x = 0 $, $ y = 1 $ (the vertex). Also verify the parabola's shape matches by testing other points like $ x = 1 $ giving $ y = 0 $.

Q: What if the vertex wasn't at (0, 1)?

You'd use the vertex form: $ y = a(x - h)^2 + k $ where (h, k) is the vertex. But since our vertex is at (0, 1), it simplifies to $ y = -x^2 + 1 $.

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 In the smiling function, the coefficient of X squared is positive

00:07 In the sad function, the coefficient of X squared is negative

00:11 Let's check the coefficient of each representation

00:14 In this case the coefficient is positive, therefore it doesn't suit the function

00:19 In this case the coefficient is negative, suitable for the function

00:24 In this case the coefficient is positive, therefore it doesn't suit the function

00:28 In this case the coefficient is positive, therefore it doesn't suit the function

00:35 Let's check the intersection point with the Y-axis

00:41 Let's substitute X=0 in the possible representation and check if the intersection point matches

00:49 The intersection points are equal, therefore the representation suits the function

00:52 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Find the corresponding algebraic representation for the function

2

Step-by-step solution

This problem involves determining the algebraic representation of a parabola that was presented graphically. Our goal is to interpret the graph and express it in terms of its equation for a downward-opening parabola.

To solve the problem, follow these steps:

Step 1: Identify the Parabola's Vertex – According to the diagram, the vertex is positioned at $(0, 1)$ , implying that at $x = 0$ , the maximum value of $y$ is 1. This indicates that the constant term $c$ in the parabola's equation will be 1.
Step 2: Determine the Parabola’s Orientation – The given parabola is described as downward-opening. This means the coefficient in front of $x^2$ must be negative. This leads us to the formula $y = -x^2 + c$ .
Step 3: Construct the Equation – With the downward orientation and the vertex point established, the equation becomes $y = -x^2 + 1$ as $c = 1$ from the vertex.

By matching one of the multiple-choice answers with our derived equation, it's clear that choice 2 corresponds to $y = -x^2 + 1$ . Thus

Therefore, the algebraic representation of the function is $y = -x^2 + 1$ .

3

Final Answer

$y=-x^2+1$

Key Points to Remember

Essential concepts to master this topic

Vertex: Identify the turning point to find the constant term
Technique: Downward parabola means negative coefficient: $y = -x^2 + 1$
Check: Substitute x = 0 to verify vertex at (0, 1) ✓

Common Mistakes

Avoid these frequent errors

Confusing parabola orientation
Don't assume upward opening just because it's a parabola = wrong sign! This gives positive coefficient when you need negative. Always check if the parabola opens up (positive coefficient) or down (negative coefficient).

Practice Quiz

Test your knowledge with interactive questions

Which chart represents the function $$ y=x^2-9 $$ ?

FAQ

Everything you need to know about this question

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

+

Look at the direction of the curve! If it opens upward like a U-shape, the coefficient of $x^2$ is positive. If it opens downward like an upside-down U, the coefficient is negative.

What does the vertex tell me about the equation?

+

The vertex gives you the constant term and the axis of symmetry. For vertex at (0, 1), the equation has +1 as the constant term, making it $y = -x^2 + 1$ .

Why is the coefficient of x² equal to -1 in this problem?

+

The graph shows a standard parabola that's flipped upside down. When there's no stretching or compressing, the coefficient is simply -1 for downward opening parabolas.

How do I verify my equation is correct?

+

Test key points! Check that when $x = 0$ , $y = 1$ (the vertex). Also verify the parabola's shape matches by testing other points like $x = 1$ giving $y = 0$ .

What if the vertex wasn't at (0, 1)?

+

You'd use the vertex form: $y = a(x - h)^2 + k$ where (h, k) is the vertex. But since our vertex is at (0, 1), it simplifies to $y = -x^2 + 1$ .

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Convert Quadratic Graph to Algebraic Form: Finding Equation for r=128.048

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