/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Graph the given functions. $$y... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 22

Graph the given functions. $$y=2+3 x+x^{2}$$

Short Answer

Expert verified
The graph is an upward-opening parabola with vertex at \((-\frac{3}{2}, \frac{-1}{4})\), y-intercept at \((0, 2)\), and x-intercepts at \((-1, 0)\) and \((-2, 0)\).

Step by step solution

01

Identify the Type of Function

The function given is in the form of a quadratic equation, which is a polynomial of degree 2. It can be identified as a parabola. The general form of a quadratic function is \( y = ax^2 + bx + c \), where \( a = 1 \), \( b = 3 \), and \( c = 2 \) in the given function.
02

Determine the Direction of the Parabola

The coefficient of \( x^2 \) is \( a = 1 \), which is positive. This means that the parabola opens upwards.
03

Find the Vertex of the Parabola

The vertex of a parabola \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = 3 \), so \( x = -\frac{3}{2 \times 1} = -\frac{3}{2} \). Substituting \( x = -\frac{3}{2} \) back into the equation gives the \( y \)-coordinate of the vertex: \( y = 2 + 3(-\frac{3}{2}) + (-\frac{3}{2})^2 = \frac{-1}{4} \). Thus, the vertex is \( (-\frac{3}{2}, \frac{-1}{4}) \).
04

Find the Y-intercept

The y-intercept of a function is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation to find \( y \): \( y = 2 + 3(0) + 0^2 = 2 \). The y-intercept is \( (0, 2) \).
05

Find the X-intercepts (If Any)

The x-intercepts are points where the graph crosses the x-axis, meaning \( y = 0 \). Set the equation equal to zero and solve for \( x \): \( 0 = x^2 + 3x + 2 \). Factoring gives \( (x + 1)(x + 2) = 0 \), so \( x = -1 \) or \( x = -2 \). These x-intercepts are \( (-1, 0) \) and \( (-2, 0) \).
06

Plot Key Points and Sketch the Graph

Plot the vertex \( (-\frac{3}{2}, \frac{-1}{4}) \), the y-intercept \( (0, 2) \), and the x-intercepts \( (-1, 0) \) and \( (-2, 0) \) on a graph. Since the parabola opens upwards, sketch a smooth curve through these points to represent the quadratic function \( y = 2 + 3x + x^2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Parabolas
Graphing a parabola is like drawing a smooth U-shaped curve on a graph. Parabolas come from quadratic functions, which typically look like this: \( y = ax^2 + bx + c \). The key to graphing them involves finding specific points: the vertex, the y-intercept, and the x-intercepts. These points give us a clear picture of where the parabola sits on the graph.
  • The **vertex** is the highest or lowest point of the parabola, depending on its direction.
  • **X-intercepts** are where the parabola crosses the x-axis.
  • **Y-intercept** is where it crosses the y-axis.
Once these points are plotted, draw a smooth curve through them to reveal the shape of the parabola. Remember, the direction of the parabola (up or down) tells you which way the arms of the U will point.
Quadratic Equation
Let's talk about quadratic equations, which are a kind of polynomial equation. They usually take the form \( y = ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) represent numbers called coefficients. For the provided equation, they are \( a = 1 \), \( b = 3 \), and \( c = 2 \).
  • The **coefficient** \( a \) determines the direction the parabola opens. Positive means up, and negative means down.
  • \( b \) and \( c \) help shape the parabola's width and position.
In our example, since \( a = 1 \) is positive, we know the parabola opens upwards like a happy face. Being familiar with these coefficients can help quickly sketch the graph of the quadratic equation.
X-intercepts
X-intercepts are the points where the parabola intersects the x-axis. To find them, set \( y = 0 \) in the quadratic equation and solve for \( x \). For the equation \( 0 = x^2 + 3x + 2 \), you can factor it to find the intercepts: \((x + 1)(x + 2) = 0\). This gives solutions \( x = -1 \) and \( x = -2 \).
  • Think of **x-intercepts** as the roots of the equation where the graph meets the x-axis.
  • These solutions give us the exact spots where the curve touches or crosses the x-axis.
Marking these intercepts on your graph helps in shaping the parabola's path properly, and they are crucial for accurate graphing.
Vertex of a Parabola
The vertex is a special point on the parabola: it's either the very top or the very bottom. You can find the x-coordinate of the vertex with the formula \( x = -\frac{b}{2a} \), and then plug that back into the equation to find the y-coordinate. For \( y = x^2 + 3x + 2 \), the vertex is \( (-\frac{3}{2}, \frac{-1}{4}) \).
  • The **vertex** gives us vital information about the maximum or minimum point of the parabola.
  • The vertex tells us where the parabola's curve changes direction.
Once the vertex is located on the graph, it helps center the parabola and guides the overall shape and path of the graph. Make sure to always plot this point carefully when graphing.

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