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Chapter 8: Problem 40

For each quadratic function, (a) find the vertex, the axis of symmetry, and the maximum or minimum function value and (b) graph the function. $$f(x)=-3 x^{2}-7 x+2$$

Short Answer

Expert verified
Vertex: \(\left(-\frac{7}{6}, 6.083\right)\), Axis of Symmetry: \(x = -\frac{7}{6}\). Maximum value: 6.083.

Step by step solution

01

Identify the Quadratic Function Form

The quadratic function is given in the standard form, which is \[ f(x) = ax^2 + bx + c \].Here, we identify that \(a = -3\), \(b = -7\), and \(c = 2\).
02

Find the Vertex

The vertex of a quadratic function in standard form can be found using the formula \( x = -\frac{b}{2a} \). Substituting \(a = -3\) and \(b = -7\), we get:\[ x = -\frac{-7}{2(-3)} = \frac{7}{-6} = -\frac{7}{6} \].To find the y-coordinate of the vertex, substitute \( x = -\frac{7}{6} \) into the function:\[ f\left(-\frac{7}{6}\right) = -3\left(-\frac{7}{6}\right)^2 - 7 \left(-\frac{7}{6}\right) + 2 \].Calculating this, \[ f\left(-\frac{7}{6}\right) = -3\left (\frac{49}{36}\right) + \left (\frac{49}{6}\right) + 2 \]. Simplifying further, \[ f(x) \approx 6.083 \].So, the vertex is \(\left(-\frac{7}{6}, 6.083\right)\).
03

Find the Axis of Symmetry

The axis of symmetry for a quadratic function is a vertical line given by the formula \( x = -\frac{b}{2a} \). From the previous step, we have this value as\( x = -\frac{7}{6} \).
04

Determine if the Vertex is a Maximum or Minimum

If the coefficient of \(x^2\), \(a\), is negative as in this function \( a = -3 \), the parabola opens downwards, and the vertex represents a maximum point. Therefore, this function has a maximum value at \( f(-\frac{7}{6}) \approx 6.083 \).
05

Graph the Quadratic Function

To graph the function, plot the vertex \( \left(-\frac{7}{6}, 6.083\right) \). Draw the axis of symmetry at \( x = -\frac{7}{6} \). Since the parabola opens downwards, sketch the parabola with the vertex being the highest point. Identify a few additional points on either side of the vertex for more accuracy.

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Key Concepts

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

vertex
The vertex of a quadratic function is a crucial point representing the peak or trough of the curve. For a quadratic equation in standard form, \[ f(x) = ax^2 + bx + c \], the vertex can be found using the formula: \[-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\].
Simply plug the coefficients into this formula. For \[f(x) = -3x^2 - 7x + 2\], \[\frac{7}{6}\] is the x-coordinate.
Substitute back into the equation to get the y-coordinate: \[-3\left(-\frac{7}{6}\right)^2 - 7\left(-\frac{7}{6}\right) + 2 = 6.083\]. So, the vertex is at \[-\frac{7}{6}\, 6.083\].
The vertex tells us either the highest or lowest point on the graph.
axis of symmetry
The axis of symmetry is a vertical line that splits the parabola into two mirror-image halves. This line passes through the vertex and shows where the function's value is symmetrical.
For the standard form quadratic equation \[f(x) = ax^2 + bx + c\], the axis of symmetry is \[x = -\frac{b}{2a}\].
In our function, \[f(x) = -3x^2 - 7x + 2\], replace into the formula to get \[x = -\frac{-7}{2\left(-3\right)} = -\frac{7}{6}\].
Drawing this vertical line on the graph helps you see the parabola's symmetry.
maximum value
The maximum value of a quadratic function occurs at the vertex if the parabola opens downwards, which happens when the coefficient \[a\] is negative.
In our equation \[f(x) = -3x^2 - 7x + 2\], \[a = -3\], so the parabola opens downwards, meaning the vertex is the highest point.
To find this maximum value, evaluate the function at the vertex's x-coordinate:
\[f\left(-\frac{7}{6}\right) = -3\left(-\frac{7}{6}\right)^2 - 7\left(-\frac{7}{6}\right) + 2 = 6.083\].
So, the maximum value is approximately \[6.083\].
graphing quadratics
Graphing quadratics starts with identifying key characteristics: vertex, axis of symmetry, and direction of the parabola (up or down).
Step-by-step, you should:
  • Plot the vertex.

  • Draw the axis of symmetry.

  • Decide if the parabola opens upwards (a > 0) or downwards (a < 0).

  • Plot additional points for accuracy on either side of the vertex.

  • Draw the curve through these points.
These steps give a precise and clear graph of the quadratic function.
standard form of quadratic equation
The standard form of a quadratic equation is \[ f(x) = ax^2 + bx + c\], where:
  • \[a\]: Determines the direction (up or down) and width of the parabola.

  • \[b\]: Influences the vertex's horizontal position.

  • \[c\]: Represents the y-intercept, or where the graph crosses the y-axis.
Understanding these components aids in sketching and interpreting quadratic functions.

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