Consider The Following Function: H ( X ) = X 2 + 4 X + 3 H(x) = X^2 + 4x + 3 H ( X ) = X 2 + 4 X + 3 Step 3 Of 4: Find Two Points On The Graph Of The Parabola Other Than The Vertex And The X-intercepts.Answer:Point A: ( □ , □ (\square, \square ( □ , □ ]Point B: ( □ , □ (\square, \square ( □ , □ ]

Introduction

In the previous steps, we have been working with the function $h(x) = x^2 + 4x + 3$. We have identified the vertex and the x-intercepts of the parabola represented by this function. However, to gain a deeper understanding of the graph, we need to find additional points that lie on the parabola. In this step, we will focus on finding two points on the graph of the parabola other than the vertex and the x-intercepts.

Understanding the Function

Before we proceed, let's take a moment to understand the function $h(x) = x^2 + 4x + 3$. This is a quadratic function, which means it represents a parabola when graphed. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In our case, $a = 1$, $b = 4$, and $c = 3$.

Finding the Vertex

The vertex of a parabola is the point on the graph where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on the direction of the parabola. To find the vertex, we can use the formula $x = -\frac{b}{2a}$. Plugging in the values of $a$ and $b$, we get $x = -\frac{4}{2(1)} = -2$. To find the y-coordinate of the vertex, we plug $x = -2$ into the function: $h(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1$. Therefore, the vertex of the parabola is at the point $(-2, -1)$.

Finding the X-Intercepts

The x-intercepts of a parabola are the points where the parabola intersects the x-axis. To find the x-intercepts, we set the function equal to zero and solve for $x$. In this case, we have $x^2 + 4x + 3 = 0$. We can factor the quadratic expression as $(x + 3)(x + 1) = 0$. Setting each factor equal to zero, we get $x + 3 = 0$ and $x + 1 = 0$. Solving for $x$, we get $x = -3$ and $x = -1$. Therefore, the x-intercepts of the parabola are at the points $(-3, 0)$ and $(-1, 0)$.

Finding Additional Points on the Graph

Now that we have identified the vertex and the x-intercepts, we need to find two additional points on the graph of the parabola. To do this, we can choose two values of $x$ that are not equal to the x-intercepts or the vertex, and plug them into the function to find the corresponding y-values.

Let's choose $x = 0$ as one of our values. Plugging $x = 0$ into the function, we get $h(0) = (0)^2 + 4(0) + 3 = 3$. Therefore, the point $(0, 3)$ lies on the graph of the parabola.

For our second value, let's choose $x = 1$. Plugging $x = 1$ into the function, we get $h(1) = (1)^2 + 4(1) + 3 = 1 + 4 + 3 = 8$. Therefore, the point $(1, 8)$ lies on the graph of the parabola.

Conclusion

In this step, we have found two additional points on the graph of the parabola represented by the function $h(x) = x^2 + 4x + 3$. These points are $(0, 3)$ and $(1, 8)$. By finding these points, we have gained a deeper understanding of the graph of the parabola and can now visualize the shape of the parabola more accurately.

Answer

Q: What is a parabola?

A: A parabola is a type of curve that is represented by a quadratic function. It is a U-shaped curve that opens upwards or downwards, depending on the direction of the parabola.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point on the graph where the parabola changes direction. It is also the minimum or maximum point of the parabola, depending on the direction of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. This formula will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you can plug the x-coordinate into the function.

Q: What are the x-intercepts of a parabola?

A: The x-intercepts of a parabola are the points where the parabola intersects the x-axis. To find the x-intercepts, you can set the function equal to zero and solve for $x$.

Q: How do I find additional points on the graph of a parabola?

A: To find additional points on the graph of a parabola, you can choose values of $x$ that are not equal to the x-intercepts or the vertex, and plug them into the function to find the corresponding y-values.

Q: What is the significance of the points $(0, 3)$ and $(1, 8)$ on the graph of the parabola?

A: The points $(0, 3)$ and $(1, 8)$ are two additional points on the graph of the parabola that we found by plugging in values of $x$ into the function. These points help to visualize the shape of the parabola and provide a deeper understanding of the graph.

Q: How can I use the graph of a parabola in real-life applications?

A: The graph of a parabola can be used in a variety of real-life applications, such as modeling the trajectory of a projectile, representing the cost of producing a product, or modeling the growth of a population.

Q: What are some common mistakes to avoid when working with parabolas?

A: Some common mistakes to avoid when working with parabolas include:

• Not using the correct formula to find the vertex or x-intercepts
• Not plugging in the correct values of $x$ into the function to find additional points on the graph
• Not checking the direction of the parabola to determine whether it is a minimum or maximum point

Q: How can I practice working with parabolas?

A: You can practice working with parabolas by:

• Graphing different parabolas and identifying their vertices and x-intercepts
• Finding additional points on the graph of a parabola by plugging in different values of $x$
• Using real-life applications to model the behavior of a parabola

Conclusion

In this article, we have explored the graph of a parabola and answered some frequently asked questions about parabolas. We have discussed the vertex and x-intercepts of a parabola, as well as how to find additional points on the graph. We have also provided some tips and common mistakes to avoid when working with parabolas. By practicing working with parabolas, you can gain a deeper understanding of the graph and its applications in real-life situations.