CURVE SKETCHING

Question

CURVE SKETCHING

𝑦 = 𝑥^2 + 4𝑥 + 3

ANSWER: 𝐶𝑃: 𝑥 = −2 , No IP, Decreasing on (−∞,−2), Increasing on (−2, ∞) , CU on (−∞, ∞) , Min point(−2,−1)

I WANT THE STEPS PLZ

Answer 1

To sketch the curve of the equation 𝑦 = 𝑥^2 + 4𝑥 + 3, you can follow these steps:

Step 1: Determine the domain
Since there are no restrictions on the value of 𝑥 in the given equation, the domain is all real numbers, or 𝑥 ∈ (-∞, ∞).

Step 2: Find the critical points
To find the critical points, take the derivative of the given equation with respect to 𝑥 and set it equal to zero. In this case, you need to find the derivative of 𝑦 = 𝑥^2 + 4𝑥 + 3.
Using the power rule and linearity of derivatives, the derivative will be 𝑑𝑦/𝑑𝑥 = 2𝑥 + 4.
Now set 2𝑥 + 4 = 0 and solve for 𝑥:
2𝑥 + 4 = 0
2𝑥 = -4
𝑥 = -4/2
𝑥 = -2.

So the critical point of the curve is 𝐶𝑃: 𝑥 = -2.

Step 3: Determine the concavity and inflection points
To find the concavity and inflection points, you need to take the second derivative of the equation. The second derivative of 𝑦 = 𝑥^2 + 4𝑥 + 3 is 𝑑^2𝑦/𝑑𝑥^2 = 2.

Since the second derivative is a constant (2), this means that the concavity is constant throughout the curve and there are no inflection points.

Step 4: Determine the increasing and decreasing intervals
To find the intervals where the curve is increasing or decreasing, you can analyze the critical points and test points in between those critical points.
Since the critical point is 𝑥 = -2, you can choose a test point in each of the intervals (-∞, -2) and (-2, ∞).

For the interval (-∞, -2), choose 𝑥 = -3. Substitute 𝑥 = -3 into the original equation 𝑦 = 𝑥^2 + 4𝑥 + 3:
𝑦 = (-3)^2 + 4(-3) + 3
𝑦 = 9 - 12 + 3
𝑦 = 0.

Since 𝑦 = 0, the curve is decreasing on the interval (-∞, -2).

For the interval (-2, ∞), choose 𝑥 = 1. Substitute 𝑥 = 1 into the original equation 𝑦 = 𝑥^2 + 4𝑥 + 3:
𝑦 = (1)^2 + 4(1) + 3
𝑦 = 1 + 4 + 3
𝑦 = 8.

Since 𝑦 = 8, the curve is increasing on the interval (-2, ∞).

Step 5: Determine the y-intercept
To find the y-intercept, substitute 𝑥 = 0 into the original equation 𝑦 = 𝑥^2 + 4𝑥 + 3:
𝑦 = (0)^2 + 4(0) + 3
𝑦 = 3.

So the y-intercept is 𝑦 = 3.

Step 6: Plot the points and sketch the curve
Based on the information gathered from the steps above, plot the points (-2,-1) as the critical point and (0, 3) as the y-intercept on a graph. Then, sketch a smooth curve that passes through these points while following the increasing and decreasing trends.

In summary:
- The critical point is 𝐶𝑃: 𝑥 = -2.
- The curve is decreasing on the interval (-∞, -2).
- The curve is increasing on the interval (-2, ∞).
- There are no inflection points.
- The y-intercept is (0, 3).
- The curve passes through the point (-2, -1) which is the minimum point.

Remember to label the axes and any other relevant details on the sketch.