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Chapter 11: Problem 21

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples I through \(4 .\) (IMAGE CANNOT COPY) \(f(x)=x^{2}+8 x+15\)

Short Answer

Expert verified
The vertex is \((-4, -1)\), the graph opens upward, and intercepts are at \((0, 15)\), \((-3, 0)\), \((-5, 0)\).

Step by step solution

01

Identify the Quadratic Function

The given quadratic function is in the standard form, which is:\[ f(x) = ax^2 + bx + c \]For the function \( f(x) = x^2 + 8x + 15 \), we have:\( a = 1 \), \( b = 8 \), and \( c = 15 \).
02

Determine the Direction the Graph Opens

The graph of a quadratic function \( ax^2 + bx + c \) opens upward if \( a > 0 \) and downward if \( a < 0 \).Since \( a = 1 \) in this function, which is greater than zero, the parabola opens upward.
03

Find the Vertex

The vertex form of a quadratic is given by:\[ x = -\frac{b}{2a} \]Substituting \( b = 8 \) and \( a = 1 \) into the formula:\[ x = -\frac{8}{2 \times 1} = -4 \]Then substitute \( x = -4 \) back into the function to find \( y \):\[ f(-4) = (-4)^2 + 8(-4) + 15 = 16 - 32 + 15 = -1 \]Thus, the vertex is \((-4, -1)\).
04

Find the Intercepts

To find the y-intercept, set \( x = 0 \):\[ f(0) = 0^2 + 8(0) + 15 = 15 \]Thus, the y-intercept is \( (0, 15) \).To find the x-intercepts, set \( f(x) = 0 \):\[ x^2 + 8x + 15 = 0 \]This can be factored into \((x+3)(x+5) = 0\), which gives:\[ x = -3 \quad \text{and} \quad x = -5 \]So the x-intercepts are \( (-3, 0) \) and \( (-5, 0) \).
05

Sketch the Graph

Using the vertex \((-4, -1)\), y-intercept \((0, 15)\), and x-intercepts \((-3, 0)\) and \((-5, 0)\), you can sketch the parabola.1. Mark the vertex \((-4, -1)\).2. Draw the axis of symmetry (vertical line through the vertex, \( x = -4 \)).3. Plot the intercepts: y-intercept \((0, 15)\), and x-intercepts \((-3, 0)\) and \((-5, 0)\).4. Draw a smooth curve through these points, ensuring it opens upwards.

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Key Concepts

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

Quadratic Function
A quadratic function is a type of polynomial function that is characterized by a degree of 2. The general form of a quadratic function is given by \( f(x) = ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). This function is pivotal in depicting a curved graph called a parabola.

Key features of a quadratic function include its vertex, axis of symmetry, and its intercepts.
  • **Vertex**: This is the turning point of the parabola and can either represent the minimum or maximum point, depending on the direction the parabola opens.
  • **Axis of Symmetry**: This is a vertical line that divides the parabola into two mirror-image halves. It's given by the equation \( x = -\frac{b}{2a} \).
  • **Intercepts**: These are the points where the graph crosses the x-axis and y-axis.
Understanding quadratic functions is crucial for solving equations and understanding the shape and properties of their graphs.
Parabola
A parabola is the graph of a quadratic function and takes on a U-shape. It is a symmetric curve that can open upwards or downwards, depending on the coefficient of the squared term in the function. For example, in the quadratic function \( f(x) = x^2 + 8x + 15 \), since \( a = 1 \), which is greater than zero, the parabola opens upwards.

Some vital aspects of a parabola include:
  • **Direction**: Determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards.
  • **Vertex**: The highest or lowest point on the parabola. For example, in this function, the vertex found is \((-4, -1)\).
  • **Symmetry**: The parabola is symmetrical about its axis of symmetry, enabling easy prediction of its shape across this axis.
A well-understood parabola is fundamental in graphing and solving real-life problems involving quadratic functions.
x-intercepts
The x-intercepts of a quadratic function are the points where the parabola crosses the x-axis. These points occur when the function value is zero \((f(x) = 0)\).

For the function \( f(x) = x^2 + 8x + 15 \), setting \( f(x) \) to zero gives the equation \( x^2 + 8x + 15 = 0 \). To find the x-intercepts:
  • Factor the quadratic expression: \((x + 3)(x + 5) = 0\).
  • Set each factor to zero: \( x + 3 = 0 \) or \( x + 5 = 0 \).
  • Solve for \( x \): \( x = -3 \) and \( x = -5 \).
Thus, the x-intercepts are \((-3, 0)\) and \((-5, 0)\). These points are crucial as they exhibit where the parabola meets the x-axis, which often corresponds to the roots of the equation in problem-solving contexts.
y-intercept
The y-intercept of a quadratic function is where the graph slices through the y-axis. This happens when the value of \( x \) is zero. In the given quadratic function \( f(x) = x^2 + 8x + 15 \), finding the y-intercept involves substituting \( x = 0 \):

\[ f(0) = 0^2 + 8(0) + 15 = 15 \]

Thus, the y-intercept is at the point \((0, 15)\).

Including the y-intercept in analyzing the graph of a quadratic function provides a complete view of its placement relative to the coordinate axes, enabling more accurate graph sketching and understanding of its behavior across the plane.

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Most popular questions from this chapter

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