When Graphing $y = 2x^2 + 35x + 75$, Which Viewing Window Would Allow You To See All Of The Intercepts And The Minimum As Closely As Possible?A. The $x$-axis From -20 To 5, And The $y$-axis From -80 To 80B. The

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Introduction

Graphing quadratic equations can be a fascinating topic in mathematics, and it's essential to understand how to visualize these equations effectively. When graphing a quadratic equation, it's crucial to choose the right viewing window to see all the intercepts and the minimum value as closely as possible. In this article, we will explore how to determine the ideal viewing window for graphing the quadratic equation $y = 2x^2 + 35x + 75$.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is $ax^2 + bx + c$, where a, b, and c are constants. The graph of a quadratic equation is a parabola, which is a U-shaped curve that opens upwards or downwards.

Finding the Intercepts

The intercepts of a quadratic equation are the points where the graph intersects the x-axis and the y-axis. To find the x-intercepts, we set y = 0 and solve for x. To find the y-intercept, we set x = 0 and solve for y.

For the given quadratic equation $y = 2x^2 + 35x + 75$, we can find the x-intercepts by setting y = 0 and solving for x:

0=2x2+35x+750 = 2x^2 + 35x + 75

Using the quadratic formula, we get:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting the values of a, b, and c, we get:

x=βˆ’35Β±352βˆ’4(2)(75)2(2)x = \frac{-35 \pm \sqrt{35^2 - 4(2)(75)}}{2(2)}

Simplifying, we get:

x=βˆ’35Β±1225βˆ’6004x = \frac{-35 \pm \sqrt{1225 - 600}}{4}

x=βˆ’35Β±6254x = \frac{-35 \pm \sqrt{625}}{4}

x=βˆ’35Β±254x = \frac{-35 \pm 25}{4}

So, the x-intercepts are:

x=βˆ’35+254=βˆ’2.5x = \frac{-35 + 25}{4} = -2.5

x=βˆ’35βˆ’254=βˆ’10.5x = \frac{-35 - 25}{4} = -10.5

To find the y-intercept, we set x = 0 and solve for y:

y=2(0)2+35(0)+75y = 2(0)^2 + 35(0) + 75

y=75y = 75

So, the y-intercept is (0, 75).

Finding the Minimum Value

The minimum value of a quadratic equation occurs at the vertex of the parabola. To find the x-coordinate of the vertex, we use the formula:

x=βˆ’b2ax = \frac{-b}{2a}

Substituting the values of a and b, we get:

x=βˆ’352(2)x = \frac{-35}{2(2)}

x=βˆ’354x = \frac{-35}{4}

x=βˆ’8.75x = -8.75

To find the y-coordinate of the vertex, we substitute the x-coordinate into the equation:

y=2(βˆ’8.75)2+35(βˆ’8.75)+75y = 2(-8.75)^2 + 35(-8.75) + 75

y=2(76.5625)βˆ’308.125+75y = 2(76.5625) - 308.125 + 75

y=153.125βˆ’308.125+75y = 153.125 - 308.125 + 75

y=βˆ’80y = -80

So, the vertex is (-8.75, -80).

Choosing the Ideal Viewing Window

To see all the intercepts and the minimum value as closely as possible, we need to choose a viewing window that includes the x-intercepts, the y-intercept, and the vertex. Based on the calculations above, we can see that the x-intercepts are -2.5 and -10.5, the y-intercept is 75, and the vertex is -8.75.

A suitable viewing window would be:

  • The x-axis from -20 to 5
  • The y-axis from -80 to 80

This viewing window includes all the intercepts and the minimum value, allowing us to visualize the graph of the quadratic equation effectively.

Conclusion

Introduction

Graphing quadratic equations can be a fascinating topic in mathematics, and it's essential to understand how to visualize these equations effectively. In our previous article, we explored how to find the intercepts and the minimum value of a quadratic equation and chose a suitable viewing window to visualize the graph. In this article, we will answer some frequently asked questions about graphing quadratic equations.

Q&A

Q: What is the general form of a quadratic equation?

A: The general form of a quadratic equation is $ax^2 + bx + c$, where a, b, and c are constants.

Q: What is the graph of a quadratic equation called?

A: The graph of a quadratic equation is a parabola, which is a U-shaped curve that opens upwards or downwards.

Q: How do I find the x-intercepts of a quadratic equation?

A: To find the x-intercepts, set y = 0 and solve for x using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I find the y-intercept of a quadratic equation?

A: To find the y-intercept, set x = 0 and solve for y.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point where the graph changes direction, and it is the minimum or maximum value of the equation.

Q: How do I find the x-coordinate of the vertex of a quadratic equation?

A: To find the x-coordinate of the vertex, use the formula: $x = \frac{-b}{2a}$

Q: How do I find the y-coordinate of the vertex of a quadratic equation?

A: To find the y-coordinate of the vertex, substitute the x-coordinate into the equation.

Q: What is the ideal viewing window for graphing a quadratic equation?

A: The ideal viewing window depends on the equation, but it should include the x-intercepts, the y-intercept, and the vertex.

Q: How do I choose the ideal viewing window for graphing a quadratic equation?

A: To choose the ideal viewing window, find the x-intercepts, the y-intercept, and the vertex of the equation and choose a window that includes all these points.

Q: What are some common mistakes to avoid when graphing quadratic equations?

A: Some common mistakes to avoid when graphing quadratic equations include:

  • Not choosing the right viewing window
  • Not finding the x-intercepts and the y-intercept
  • Not finding the vertex of the equation
  • Not using the correct formula to find the x-coordinate and y-coordinate of the vertex

Conclusion

In conclusion, graphing quadratic equations can be a fascinating topic in mathematics, and it's essential to understand how to visualize these equations effectively. By understanding the intercepts and the minimum value of the equation, we can determine the ideal viewing window to see all the key features of the graph. In this article, we answered some frequently asked questions about graphing quadratic equations and provided some tips to avoid common mistakes.

Additional Resources

Final Thoughts

Graphing quadratic equations is an essential skill in mathematics, and it's crucial to understand how to visualize these equations effectively. By following the tips and guidelines provided in this article, you can become proficient in graphing quadratic equations and solve a wide range of problems. Remember to always choose the right viewing window, find the x-intercepts and the y-intercept, and find the vertex of the equation to visualize the graph effectively.