Find the Algebraic Equation: Quadratic Function with Radius 6

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

Look at the shape! If it looks like a smile (∪), it opens upward with positive coefficient. If it looks like a frown (∩), it opens downward with negative coefficient.

Q: What does the number 6 represent in the graph?

The number 6 is the y-intercept - where the parabola crosses the y-axis. This becomes the constant term in your equation: $ y = -x^2 + 6 $.

Q: Why isn't the equation y = x² + 6?

Because the parabola opens downward, not upward! The equation $ y = x^2 + 6 $ would create an upward-opening parabola, which is the opposite of what's shown.

Q: How can I verify my equation is correct?

Test key points! At $ x = 0 $: $ y = -0^2 + 6 = 6 $ ✓. The vertex should be at (0,6) and the parabola should open downward, matching the graph.

Q: What if I see a different radius or y-intercept?

The same method applies! The y-intercept becomes your constant term, and the parabola direction determines if your $ x^2 $ coefficient is positive or negative.

Parabola Identification with Downward Opening

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

00:15 Let's choose the right algebra expression for the function.

00:20 In a sad function, the X squared term has a negative number in front.

00:25 But in a smiling function, the X squared term has a positive number.

00:30 Here, the number is positive, so it doesn't match the sad function.

00:37 Here, the number is negative. So, it fits!

00:42 And here, the number is positive again, so it doesn't fit.

00:47 In this case too, the positive number means it doesn't fit.

00:52 And that's the solution to our question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the corresponding algebraic representation for the function

Step-by-step solution

To solve this problem, we need to identify the algebraic representation of the given graph of a function. Based on the observation of the parabola, which opens downward and intersects the y-axis at the point $y = 6$ , we can deduce its form.

Step 1: Identify the type of parabola.
The graph shows a parabola opening downwards, indicating that the leading coefficient $a$ in the equation $y = ax^2 + c$ is negative.

Step 2: Identify key points on the graph.
Since the parabola intersects the y-axis at $y = 6$ , this means when $x = 0$ , $y = 6$ . Thus, the equation $y = ax^2 + c$ simplifies to $y = -x^2 + 6$ .

Step 3: Confirmation of the algebraic representation.
From the downward orientation and the vertical intersection at $y = 6$ without any lateral shifts, the equation $y = -x^2 + 6$ fully describes the parabola.

Therefore, the corresponding algebraic representation of the function is $y = -x^2 + 6$ .

Final Answer

$y=-x^2+6$

Key Points to Remember

Essential concepts to master this topic

Direction: Downward opening parabola means negative coefficient for $x^2$
Y-intercept: Where graph crosses y-axis gives constant term: y = 6 when x = 0
Verification: Check vertex and direction match equation $y = -x^2 + 6$ ✓

Common Mistakes

Avoid these frequent errors

Confusing parabola direction with coefficient sign
Don't assume upward opening = positive coefficient when the graph clearly opens downward = wrong equation like $y = x^2 + 6$ ! This gives opposite direction. Always match parabola direction: downward opening means negative $x^2$ 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 do I know if the parabola opens up or down?

Look at the shape! If it looks like a smile (∪), it opens upward with positive coefficient. If it looks like a frown (∩), it opens downward with negative coefficient.

What does the number 6 represent in the graph?

The number 6 is the y-intercept - where the parabola crosses the y-axis. This becomes the constant term in your equation: $y = -x^2 + 6$ .

Why isn't the equation y = x² + 6?

Because the parabola opens downward, not upward! The equation $y = x^2 + 6$ would create an upward-opening parabola, which is the opposite of what's shown.

How can I verify my equation is correct?

Test key points! At $x = 0$ : $y = -0^2 + 6 = 6$ ✓. The vertex should be at (0,6) and the parabola should open downward, matching the graph.

What if I see a different radius or y-intercept?

The same method applies! The y-intercept becomes your constant term, and the parabola direction determines if your $x^2$ coefficient is positive or negative.

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