The $x$-intercepts Occur Where $x = -3$ And $x = 8$.Part B:Draw A Diagram Of The Archway Modeled By The Equation $y = -x^2 + 5x + 24$. - Find And Label The $y$-intercept And The

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The Art of Graphing: Understanding the $y = -x^2 + 5x + 24$ Equation

In mathematics, graphing is an essential skill that helps us visualize and understand the behavior of various functions. One such function is the quadratic equation, which is represented by the equation $y = -x^2 + 5x + 24$. In this article, we will delve into the world of graphing and explore the properties of this quadratic equation. We will also discuss how to find and label the $y$-intercept and the $x$-intercepts, which are crucial components of a graph.

The given equation is a quadratic equation in the form of $y = ax^2 + bx + c$, where $a = -1$, $b = 5$, and $c = 24$. The coefficient $a$ determines the direction of the parabola's opening, while the coefficients $b$ and $c$ affect the parabola's position and shape.

The $y$-intercept is the point where the graph intersects the $y$-axis. To find the $y$-intercept, we set $x = 0$ and solve for $y$. Substituting $x = 0$ into the equation, we get:

y=−02+5(0)+24y = -0^2 + 5(0) + 24

y=24y = 24

Therefore, the $y$-intercept is $(0, 24)$.

The $x$-intercepts are the points where the graph intersects the $x$-axis. To find the $x$-intercepts, we set $y = 0$ and solve for $x$. Substituting $y = 0$ into the equation, we get:

0=−x2+5x+240 = -x^2 + 5x + 24

We can solve this quadratic equation using the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

x=−5±52−4(−1)(24)2(−1)x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(24)}}{2(-1)}

x=−5±25+96−2x = \frac{-5 \pm \sqrt{25 + 96}}{-2}

x=−5±121−2x = \frac{-5 \pm \sqrt{121}}{-2}

x=−5±11−2x = \frac{-5 \pm 11}{-2}

Therefore, the $x$-intercepts are $x = -3$ and $x = 8$.

Now that we have found the $y$-intercept and the $x$-intercepts, we can draw the graph of the equation. The graph will be a parabola that opens downward, since the coefficient $a$ is negative.

To draw the graph, we can start by plotting the $y$-intercept at $(0, 24)$. Then, we can plot the $x$-intercepts at $(-3, 0)$ and $(8, 0)$. Finally, we can draw a smooth curve through these points to form the parabola.

In conclusion, graphing the equation $y = -x^2 + 5x + 24$ requires finding the $y$-intercept and the $x$-intercepts. We can find the $y$-intercept by setting $x = 0$ and solving for $y$, and we can find the $x$-intercepts by setting $y = 0$ and solving for $x$. By plotting these points and drawing a smooth curve through them, we can create a graph that accurately represents the equation.

  • What are some real-world applications of graphing quadratic equations?
  • How can we use graphing to solve problems in physics and engineering?
  • What are some common mistakes to avoid when graphing quadratic equations?
  • Some real-world applications of graphing quadratic equations include modeling the trajectory of a projectile, representing the growth of a population, and analyzing the motion of an object.
  • Graphing can be used to solve problems in physics and engineering by modeling the behavior of complex systems and predicting the outcomes of different scenarios.
  • Some common mistakes to avoid when graphing quadratic equations include:
    • Failing to find the $y$-intercept and the $x$-intercepts.
    • Plotting the graph incorrectly, such as drawing a parabola that opens upward when the coefficient $a$ is negative.
    • Not using a smooth curve to connect the points on the graph.
  • For more information on graphing quadratic equations, see the following resources:
    • Khan Academy: Graphing Quadratic Equations
    • Mathway: Graphing Quadratic Equations
    • Wolfram Alpha: Graphing Quadratic Equations
      Frequently Asked Questions: Graphing Quadratic Equations

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. It is typically written in the form of ax + b = 0, where a and b are constants. Quadratic equations, on the other hand, have a higher degree and are typically written in the form of ax^2 + bx + c = 0.

A: To graph a quadratic equation, you need to find the y-intercept and the x-intercepts. The y-intercept is the point where the graph intersects the y-axis, and the x-intercepts are the points where the graph intersects the x-axis. You can find the y-intercept by setting x = 0 and solving for y, and you can find the x-intercepts by setting y = 0 and solving for x.

A: The y-intercept is the point where the graph intersects the y-axis. It is the value of y when x = 0.

A: The x-intercept is the point where the graph intersects the x-axis. It is the value of x when y = 0.

A: To find the x-intercepts of a quadratic equation, you need to set y = 0 and solve for x. You can use the quadratic formula to solve for x: x = (-b ± √(b^2 - 4ac)) / 2a.

A: The quadratic formula is a formula used to solve quadratic equations. It is written as x = (-b ± √(b^2 - 4ac)) / 2a.

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, you need to simplify the expression and solve for x.

A: The x-intercepts of the equation y = -x^2 + 5x + 24 are x = -3 and x = 8.

A: The y-intercept of the equation y = -x^2 + 5x + 24 is y = 24.

A: To graph the equation y = -x^2 + 5x + 24, you need to find the y-intercept and the x-intercepts. Then, you can plot the points and draw a smooth curve through them to form the graph.

A: Some real-world applications of graphing quadratic equations include modeling the trajectory of a projectile, representing the growth of a population, and analyzing the motion of an object.

A: You can use graphing to solve problems in physics and engineering by modeling the behavior of complex systems and predicting the outcomes of different scenarios.

A: Some common mistakes to avoid when graphing quadratic equations include failing to find the y-intercept and the x-intercepts, plotting the graph incorrectly, and not using a smooth curve to connect the points on the graph.