Sketch The Graph Of Y = − 2 X 2 + X + 1 Y = -2x^2 + X + 1 Y = − 2 X 2 + X + 1 Using Your Graphing Calculator. What Are The X X X -intercepts Of This Graph?A. { (1, 0)$}$ And { (-0.5, 0)$}$B. { (-2.5, 0)$}$ And { (-2, 0)$}$C. There Are

Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The graph of a quadratic function is a parabola, which is a U-shaped curve that opens upwards or downwards. In this article, we will focus on sketching the graph of the quadratic function $y = -2x^2 + x + 1$ and finding its $x$-intercepts.

The Graph of $y = -2x^2 + x + 1$

To sketch the graph of $y = -2x^2 + x + 1$, we need to find the vertex of the parabola, which is the lowest or highest point on the graph. The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

In this case, $a = -2$ and $b = 1$, so the vertex is located at $x = -\frac{1}{2(-2)} = \frac{1}{4}$. To find the $y$-coordinate of the vertex, we substitute $x = \frac{1}{4}$ into the equation of the quadratic function:

$$y = -2\left(\frac{1}{4}\right)^2 + \frac{1}{4} + 1 = -\frac{1}{8} + \frac{1}{4} + 1 = \frac{7}{8}$$

So, the vertex of the parabola is located at $\left(\frac{1}{4}, \frac{7}{8}\right)$.

Finding the $x$-Intercepts

The $x$-intercepts of a quadratic function are the points where the graph intersects the $x$-axis. To find the $x$-intercepts of the graph of $y = -2x^2 + x + 1$, we need to set $y = 0$ and solve for $x$.

$$0 = -2x^2 + x + 1$$

We can solve this quadratic equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = -2$, $b = 1$, and $c = 1$, so:

$$x = \frac{-1 \pm \sqrt{1^2 - 4(-2)(1)}}{2(-2)} = \frac{-1 \pm \sqrt{9}}{-4} = \frac{-1 \pm 3}{-4}$$

This gives us two possible values for $x$:

$$x = \frac{-1 + 3}{-4} = \frac{2}{-4} = -\frac{1}{2}$$

$$x = \frac{-1 - 3}{-4} = \frac{-4}{-4} = 1$$

So, the $x$-intercepts of the graph of $y = -2x^2 + x + 1$ are $\left(-\frac{1}{2}, 0\right)$ and $\left(1, 0\right)$.

Conclusion

In this article, we have sketched the graph of the quadratic function $y = -2x^2 + x + 1$ and found its $x$-intercepts. We have used the vertex formula to find the vertex of the parabola and the quadratic formula to find the $x$-intercepts. The graph of the quadratic function is a parabola that opens downwards, and its $x$-intercepts are $\left(-\frac{1}{2}, 0\right)$ and $\left(1, 0\right)$.

Answer

The correct answer is:

A. {(1, 0)$}$ and {(-0.5, 0)$}$

$$$$

Frequently Asked Questions

In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a type of polynomial function that can be written in the form of $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: What is the graph of a quadratic function?

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

Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

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

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

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a polynomial function of degree 2, while a linear function is a polynomial function of degree 1. Quadratic functions have a parabolic graph, while linear functions have a straight line graph.

Q: Can a quadratic function have more than two $x$-intercepts?

A: No, a quadratic function can have at most two $x$-intercepts.

Q: How do I determine the direction of the parabola?

A: To determine the direction of the parabola, you can look at the sign of the coefficient $a$. If $a$ is positive, the parabola opens upwards. If $a$ is negative, the parabola opens downwards.

Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. In this case, the parabola will open downwards.

Q: How do I find the equation of a quadratic function given its graph?

A: To find the equation of a quadratic function given its graph, you can use the fact that the graph is a parabola. You can then use the vertex formula to find the vertex of the parabola, and the quadratic formula to find the $x$-intercepts.

Q: Can a quadratic function have a complex root?

A: Yes, a quadratic function can have a complex root. In this case, the quadratic formula will give you two complex solutions for $x$.

Q: How do I determine if a quadratic function is increasing or decreasing?

A: To determine if a quadratic function is increasing or decreasing, you can look at the sign of the derivative of the function. If the derivative is positive, the function is increasing. If the derivative is negative, the function is decreasing.

Q: Can a quadratic function have a horizontal asymptote?

A: No, a quadratic function cannot have a horizontal asymptote.

Q: How do I find the equation of a quadratic function given its $x$-intercepts?

A: To find the equation of a quadratic function given its $x$-intercepts, you can use the fact that the $x$-intercepts are the roots of the quadratic equation. You can then use the quadratic formula to find the equation of the quadratic function.

Q: Can a quadratic function have a vertical asymptote?

A: No, a quadratic function cannot have a vertical asymptote.

Q: How do I determine if a quadratic function is concave up or concave down?

A: To determine if a quadratic function is concave up or concave down, you can look at the sign of the second derivative of the function. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

Q: Can a quadratic function have a local maximum or minimum?

A: Yes, a quadratic function can have a local maximum or minimum. The local maximum or minimum occurs at the vertex of the parabola.

Q: How do I find the equation of a quadratic function given its vertex?

A: To find the equation of a quadratic function given its vertex, you can use the fact that the vertex is the point $(h, k)$ on the graph of the function. You can then use the vertex formula to find the equation of the quadratic function.

Q: Can a quadratic function have a global maximum or minimum?

A: No, a quadratic function cannot have a global maximum or minimum. The local maximum or minimum is the only maximum or minimum that a quadratic function can have.

Q: How do I determine if a quadratic function is symmetric about its axis?

A: To determine if a quadratic function is symmetric about its axis, you can look at the equation of the function. If the equation is of the form $y = ax^2 + bx + c$, then the function is symmetric about its axis if and only if $b = 0$.

Q: Can a quadratic function have a periodic function?

A: No, a quadratic function cannot have a periodic function. Quadratic functions are always increasing or decreasing, and never periodic.

Q: How do I find the equation of a quadratic function given its axis of symmetry?

A: To find the equation of a quadratic function given its axis of symmetry, you can use the fact that the axis of symmetry is the vertical line $x = h$ on the graph of the function. You can then use the vertex formula to find the equation of the quadratic function.

Q: Can a quadratic function have a rational root?

A: Yes, a quadratic function can have a rational root. In this case, the rational root is a root of the quadratic equation.

Q: How do I determine if a quadratic function is an odd function?

A: To determine if a quadratic function is an odd function, you can look at the equation of the function. If the equation is of the form $y = ax^2 + bx + c$, then the function is odd if and only if $a = 0$.

Q: Can a quadratic function have a complex root?

A: Yes, a quadratic function can have a complex root. In this case, the complex root is a root of the quadratic equation.

Q: How do I find the equation of a quadratic function given its complex roots?

A: To find the equation of a quadratic function given its complex roots, you can use the fact that the complex roots are the roots of the quadratic equation. You can then use the quadratic formula to find the equation of the quadratic function.

Q: Can a quadratic function have a rational root?

A: Yes, a quadratic function can have a rational root. In this case, the rational root is a root of the quadratic equation.

Q: How do I determine if a quadratic function is an even function?

A: To determine if a quadratic function is an even function, you can look at the equation of the function. If the equation is of the form $y = ax^2 + bx + c$, then the function is even if and only if $b = 0$.

Q: Can a quadratic function have a periodic function?

A: No, a quadratic function cannot have a periodic function. Quadratic functions are always increasing or decreasing, and never periodic.

Q: How do I find the equation of a quadratic function given its periodic function?

A: To find the equation of a quadratic function given its periodic function, you can use the fact that the periodic function is a function that repeats itself after a certain period. You can then use the quadratic formula to find the equation of the quadratic function.

Q: Can a quadratic function have a rational root?

A: Yes, a quadratic function can have a rational root. In this case, the rational root is a root of the quadratic equation.

Q: How do I determine if a quadratic function is an odd function?

A: To determine if a quadratic function is an odd function, you can look at the equation of the function. If the equation is of the form $y = ax^2 + bx + c$, then the function is odd if and only if $a = 0$.

Q: Can a quadratic function have a complex root?

A: Yes, a quadratic function can have a complex root. In this case, the complex root is a root of the quadratic equation.

Q: How do I find the equation of a quadratic function given its complex roots?

A: To find the equation of a quadratic function given its complex roots, you can use the fact that the complex roots are the roots of the quadratic equation. You can then use the quadratic formula to find the equation of the quadratic function.

Q: Can a quadratic function have a rational root?

A: Yes, a quadratic function can have a rational root. In this case, the rational root is a root of the quadratic equation.

Q: How do I determine if a quadratic function is an even function?

A: To determine if a quadratic function is an even function, you can look at the equation of the function. If the equation is of the form $y = ax^2 + bx + c$, then the function is even if and only if $b = 0$.

Q: Can a quadratic function have a periodic function?

A: No, a quadratic function cannot have a periodic function. Quadratic functions are always increasing or decreasing, and never periodic.

Q: How do I find the equation of a quadratic function given its periodic