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Chapter 8: Problem 28

For each quadratic function, (a) find the vertex and the axis of symmetry and (b) graph the function. $$g(x)=x^{2}+5 x+4$$

Short Answer

Expert verified
Vertex: (-2.5, -2.25), Axis of symmetry: x = -2.5.

Step by step solution

01

Identify coefficients

Identify the coefficients from the quadratic function in standard form, which is of the form \( ax^2 + bx + c \). Here, \( a = 1 \), \( b = 5 \), and \( c = 4 \).
02

Calculate the vertex

The vertex of a quadratic function \( ax^2 + bx + c \) is given by \( \left( -\frac{b}{2a}, g\left( -\frac{b}{2a} \right) \right) \). Calculate \( x \) for the vertex: \( x = -\frac{5}{2 \times 1} = -2.5 \). Substitute \( x = -2.5 \) back into the function to find \( g(-2.5) \): \[ g(-2.5) = (-2.5)^2 + 5(-2.5) + 4 = 6.25 - 12.5 + 4 = -2.25 \] Therefore, the vertex is \( (-2.5, -2.25) \).
03

Determine the axis of symmetry

The axis of symmetry of a quadratic function \( ax^2 + bx + c \) is the vertical line \( x = -\frac{b}{2a} \). From Step 2, we already calculated \( x = -2.5 \). Thus, the axis of symmetry is \( x = -2.5 \).
04

Graph the function

To graph the function, plot the vertex \( (-2.5, -2.25) \) on the coordinate plane, and draw the axis of symmetry at \( x = -2.5 \). Then, find and plot additional points by choosing values for \( x \) and computing corresponding \( g(x) \). For example: - For \( x = -3 \): \( g(-3) = (-3)^2 + 5(-3) + 4 = 9 - 15 + 4 = -2 \) - For \( x = -1 \): \( g(-1) = (-1)^2 + 5(-1) + 4 = 1 - 5 + 4 = 0 \) Connect these points with a smooth curve that opens upwards because the coefficient \( a = 1 \) is positive.

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Key Concepts

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

Vertex of a Quadratic Function
In a quadratic function of the form \( ax^2 + bx + c \), the vertex is a crucial point because it represents the maximum or minimum of the function. The vertex can be calculated using the formula: \( \text{Vertex} = \bigg( -\frac{b}{2a}, g\left( -\frac{b}{2a} \right) \bigg) \). Here, you need to compute both the x-coordinate and y-coordinate. For instance, in the function \( g(x) = x^2 + 5x + 4 \), we follow these steps:
  • Calculate the x-coordinate: \( x = -\frac{5}{2 \times 1} = -2.5 \).
  • Substitute \( x = -2.5 \) back into the function to find the y-coordinate: \( g(-2.5) = (-2.5)^2 + 5(-2.5) + 4 = -2.25 \).
So, the vertex is \( (-2.5, -2.25) \).
Axis of Symmetry
The axis of symmetry in a quadratic function is a vertical line that passes through the vertex, splitting the parabola into two mirror-image halves. The formula for finding the axis of symmetry is very straightforward: \( x = -\frac{b}{2a} \). This is essentially the x-coordinate of the vertex. Using our example function \( g(x) = x^2 + 5x + 4 \), and already knowing \( x = -2.5 \) from the vertex calculation, we can say the axis of symmetry is:

x = -2.5

This line is crucial for graphing, as it helps you plot the parabola accurately by knowing where it will be mirrored.
Graphing Quadratic Functions
Graphing a quadratic function involves plotting points accurately and drawing a smooth curve (parabola). Here鈥檚 a simple plan to follow:
  • First, plot the vertex. For \( g(x) = x^2 + 5x + 4 \), the vertex is at \( (-2.5, -2.25) \).
  • Draw the axis of symmetry, which is a vertical line at \( x = -2.5 \).
  • Find additional points by inserting values for x and calculating g(x). For example, at \( x = -3 \) and \( x = -1 \), the corresponding points are \( (-3, -2) \) and \( (-1, 0) \).
Once you have enough points and the axis of symmetry, connect these points with a smooth curve that opens upwards (since the coefficient \( a \) is positive). This visualization is key to understanding the function's behavior.
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is \( ax^2 + bx + c \). Here, \( a \), \( b \), and \( c \) are constants, where:\
  • \( a \) determines the parabola's direction (upward if positive, downward if negative).
  • \( b \) affects the vertex鈥檚 x-coordinate and the line of symmetry.
  • \( c \) represents the y-intercept, the point where the parabola crosses the y-axis.
In the function \( g(x) = x^2 + 5x + 4 \), we identify \( a = 1 \), \( b = 5 \), and \( c = 4 \). Understanding these parameters helps in plotting and analyzing the quadratic graph.

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