Find the Domain of y=-3⅓x²-0.4: Complete Function Analysis

Q: What's the difference between domain and range for this function?

The domain is all possible x-values (input), which is all real numbers. The range is all possible y-values (output), which are all negative numbers since the parabola opens downward and stays below the x-axis.

Q: How do I convert the mixed number -3⅓?

Convert to improper fraction: $ -3\frac{1}{3} = -\frac{10}{3} $. Multiply the whole number by denominator, add numerator: 3 × 3 + 1 = 10, so $ -\frac{10}{3} $.

Q: Why doesn't this parabola cross the x-axis?

Setting y = 0 gives $ x^2 = -0.12 $, which is negative. Since x² cannot be negative for real numbers, there are no x-intercepts - the parabola stays entirely below the x-axis.

Q: Does the negative coefficient affect the domain?

No! The negative coefficient $ -\frac{10}{3} $ only affects the parabola's direction (opens downward) and the range of y-values. The domain is still all real numbers for any quadratic function.

Quadratic Function Domains with Negative Leading Coefficients

Find the positive and negative domains of the following function:

$y=-3\frac{1}{3}x^2-0.4$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive and negative domains of the following function:

$y=-3\frac{1}{3}x^2-0.4$

Step-by-step solution

To solve this problem, we examine the function $y = -\frac{10}{3}x^2 - 0.4$ .

Step 1: Identify $a = -\frac{10}{3}$ , which is negative. This means the parabola opens downwards.

Step 2: Find the zeros by setting $y = 0$ :

$-\frac{10}{3}x^2 - 0.4 = 0$ . Solving gives $-\frac{10}{3}x^2 = 0.4$ , yielding

$x^2 = -\frac{0.4 \times 3}{10}$ , which is negative. Therefore, no real solutions exist for zero crossing.

Step 3: Recognize since it doesn't cross the x-axis, the entire parabola is below the x-axis.

Conclusion: For $x < 0$ , $y$ is negative for all $x$ ; for $x > 0$ , $y$ remains negative.

Therefore, the solution is:
$x < 0 :$ all $x$
$x > 0 :$ none

Final Answer

$x < 0 :$ all $x$

$x > 0 :$ none

Key Points to Remember

Essential concepts to master this topic

Rule: Quadratic functions have domain of all real numbers x
Technique: Mixed number -3⅓ equals improper fraction $-\frac{10}{3}$
Check: Verify parabola opens downward since a = $-\frac{10}{3} < 0$ ✓

Common Mistakes

Avoid these frequent errors

Confusing domain with range of negative function values
Don't think negative y-values mean restricted domain = wrong answer! The function produces negative outputs, but domain is still all real x-values. Always remember domain asks "what x-values work" not "what y-values result".

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

Find all values of $$ x $$ where $f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

What's the difference between domain and range for this function?

The domain is all possible x-values (input), which is all real numbers. The range is all possible y-values (output), which are all negative numbers since the parabola opens downward and stays below the x-axis.

Why does the question ask about positive and negative domains?

This is asking which parts of the domain (x < 0 or x > 0) are valid. Since quadratic functions accept all real numbers, both negative and positive x-values work - the domain includes everything!

How do I convert the mixed number -3⅓?

Convert to improper fraction: $-3\frac{1}{3} = -\frac{10}{3}$ . Multiply the whole number by denominator, add numerator: 3 × 3 + 1 = 10, so $-\frac{10}{3}$ .

Why doesn't this parabola cross the x-axis?

Setting y = 0 gives $x^2 = -0.12$ , which is negative. Since x² cannot be negative for real numbers, there are no x-intercepts - the parabola stays entirely below the x-axis.

Does the negative coefficient affect the domain?

No! The negative coefficient $-\frac{10}{3}$ only affects the parabola's direction (opens downward) and the range of y-values. The domain is still all real numbers for any quadratic function.

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