Math  /  Algebra

QuestionGraph the inequality. x2+y>3x^{2}+y>3

Studdy Solution

STEP 1

What is this asking? We need to shade the region above the parabola x2+y=3x^2 + y = 3. Watch out! Don't forget to use a dashed line for the parabola since the inequality is strict (greater than, not greater than or equal to).

STEP 2

1. Rewrite the inequality
2. Plot the parabola
3. Shade the correct region

STEP 3

To make it easier to graph, let's **rewrite** the inequality by isolating yy: x2+y>3x^2 + y > 3 Subtract x2x^2 from both sides of the inequality: y>3x2y > 3 - x^2 Now we have yy by itself!

STEP 4

To graph the **boundary** of our inequality, we'll consider the corresponding equation: y=3x2y = 3 - x^2 This is a **parabola** that opens downwards, shifted **3 units up**.

STEP 5

The **vertex** of the parabola is at (0,3)(0, 3).

STEP 6

To find the x-intercepts, set y=0y = 0: 0=3x20 = 3 - x^2 x2=3x^2 = 3x=±3x = \pm\sqrt{3}So, the **x-intercepts** are at (3,0)(-\sqrt{3}, 0) and (3,0)(\sqrt{3}, 0).

STEP 7

Now, we can **sketch** the parabola.
Remember to use a **dashed line** because the inequality is y>3x2y > 3 - x^2, not y3x2y \ge 3 - x^2.

STEP 8

Let's **test** the point (0,4)(0, 4), which is above the parabola, in our inequality y>3x2y > 3 - x^2: 4>3024 > 3 - 0^2 4>34 > 3This is **true**!

STEP 9

Since the test point (0,4)(0, 4) above the parabola satisfies the inequality, we **shade the region above** the dashed parabola.

STEP 10

The solution is the shaded region above the dashed parabola y=3x2y = 3 - x^2.
This represents all the points (x,y)(x, y) that satisfy the inequality x2+y>3x^2 + y > 3.

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