Find the Decreasing Domain: Analyzing y = -x² + 2x + 35

Q: How do I know which direction the parabola opens?

Look at the coefficient of $ x^2 $! If it's negative (like -1 in our function), the parabola opens downward. If it's positive, it opens upward.

Q: Why does the function decrease after x = 1?

Since our parabola opens downward, it reaches its highest point at the vertex (x = 1). After that peak, the function values get smaller as x increases, meaning it's decreasing.

Q: What's the difference between domain of decrease and just domain?

The domain is all possible x-values (here: all real numbers). The domain of decrease is only the x-values where the function is getting smaller as x increases.

Q: How can I verify my answer without graphing?

Pick test points! Choose values like x = 0, x = 1, and x = 2. Calculate $ y $ for each. If y-values get smaller as x increases past the vertex, you've found the decreasing interval!

Q: Does every quadratic have a decreasing interval?

Yes! Every parabola has both increasing and decreasing intervals, separated by the vertex. The direction depends on whether the parabola opens upward or downward.

Quadratic Functions with Decreasing Intervals

Find the domain of decrease of the function:

$y=-x^2+2x+35$

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Understand the problem

Find the domain of decrease of the function:

$y=-x^2+2x+35$

Step-by-step solution

To determine the domain over which the quadratic function $y = -x^2 + 2x + 35$ is decreasing, we proceed by identifying the vertex of the parabola.

Given the form $y = ax^2 + bx + c$ , we have $a = -1$ , $b = 2$ , and $c = 35$ . The x-coordinate of the vertex can be found using the formula:

x = -\frac{b}{2a}

Substituting $b = 2$ and $a = -1$ into the formula, we calculate:

x = -\frac{2}{2 \times (-1)} = -\frac{2}{-2} = 1

The vertex of the parabola occurs at $x = 1$ . Since the function is a downward-opening parabola (as indicated by the negative coefficient of $-x^2$ ), the function decreases for all $x$ values greater than the x-coordinate of the vertex.

Therefore, the domain of decrease for the function is $x > 1$ .

This matches the answer choice:

$x > 1$

Final Answer

$x > 1$

Key Points to Remember

Essential concepts to master this topic

Vertex Rule: For parabola y = ax² + bx + c, vertex x-coordinate is -b/2a
Direction: When a < 0, parabola opens downward and decreases after vertex x = 1
Check: Test points: f(0) = 35, f(2) = 31, so 35 > 31 confirms decreasing ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing and decreasing intervals
Don't say the function decreases where x < 1 because the parabola opens downward = wrong direction! This ignores that downward parabolas increase before the vertex and decrease after. Always remember: negative coefficient means decrease to the RIGHT of vertex.

Practice Quiz

Test your knowledge with interactive questions

Note that the graph of the function shown below does not intersect the x-axis

The parabola's vertex is A

Identify the interval where the function is decreasing:

FAQ

Everything you need to know about this question

How do I know which direction the parabola opens?

Look at the coefficient of $x^2$ ! If it's negative (like -1 in our function), the parabola opens downward. If it's positive, it opens upward.

Why does the function decrease after x = 1?

Since our parabola opens downward, it reaches its highest point at the vertex (x = 1). After that peak, the function values get smaller as x increases, meaning it's decreasing.

What's the difference between domain of decrease and just domain?

The domain is all possible x-values (here: all real numbers). The domain of decrease is only the x-values where the function is getting smaller as x increases.

How can I verify my answer without graphing?

Pick test points! Choose values like x = 0, x = 1, and x = 2. Calculate $y$ for each. If y-values get smaller as x increases past the vertex, you've found the decreasing interval!

Does every quadratic have a decreasing interval?

Yes! Every parabola has both increasing and decreasing intervals, separated by the vertex. The direction depends on whether the parabola opens upward or downward.

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Find the Decreasing Domain: Analyzing y = -x² + 2x + 35

Quadratic Functions with Decreasing Intervals

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which direction the parabola opens?

Why does the function decrease after x = 1?

What's the difference between domain of decrease and just domain?

How can I verify my answer without graphing?

Does every quadratic have a decreasing interval?

🌟 Unlock Your Math Potential

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