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Chapter 9: Problem 98

Sketch the graph of the function. Label the vertex. y=-3 x^{2}-5 x+3

Short Answer

Expert verified
The vertex of the function \(y=-3x^{2}-5x+3\) is approximately (5/6, -1.42).

Step by step solution

01

Identify the coefficients

The given function is \(y=-3x^{2}-5x+3\). Here, \(a=-3\), \(b=-5\), and \(c=3\).
02

Use the vertex formula to find the x-coordinate of the vertex

The x-coordinate of the vertex is given by \(-b/2a\). So, substitute \(a=-3\) and \(b=-5\). Calculating gives \(x=-(-5)/2(-3) = 5/6\).
03

Substitute the x-coordinate into the function to find the y-coordinate

Now substitute \(x=5/6\) into the function to find the y-coordinate of the vertex. \(y= -3 (5/6)^{2}-5(5/6)+3 \). Computing this, we find y equals approximately -1.42.
04

Plot the graph

Now, based on the x and y coordinates obtained for the vertex and given function, sketch the graph. Plot the point (5/6, -1.42) and label it as the vertex. Because \(a < 0\), the parabola opens downward.

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Key Concepts

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

Vertex Form
The vertex form of a quadratic function is a streamlined way to express the function making it easy to identify the vertex of the parabola. It is presented as \( y = a(x - h)^2 + k \), where
  • \(a\) represents the same coefficient as in standard or general form, indicating whether the parabola opens up or down.
  • \(h\) and \(k\) will give you the vertex point \((h, k)\).
The vertex form is particularly useful when you are given the vertex or need to graph the function quickly, as it emphasizes the vertex's location. While the given equation \( y = -3x^2 - 5x + 3 \) is not in vertex form, understanding how to manipulate the equation into this form through completing the square can give us an explicit view of its vertex and shape.
Parabola
In mathematics, a parabola is a curve where any point is at an equal distance from a fixed point known as the focus and a fixed straight line called the directrix. For quadratic functions like \( y = -3x^2 - 5x + 3 \), the graph of the equation is a parabola.
  • The orientation of a parabola is determined by the sign of the coefficient \(a\) in the standard form \( y = ax^2 + bx + c \).
  • If \(a\) is negative, as in our case where \(a = -3\), the parabola opens downward.
  • This downward opening indicates that the vertex of the parabola is a maximum point.
Understanding these properties allows you to predict and graph the curve's shape and direction.
Graphing Functions
Graphing a function means translating the algebraic form into a visually interpretable graph. For quadratic functions, graphing includes steps like identifying the vertex and symmetry, plotting key points, and determining the direction of the curve.
  • Locate the vertex using the vertex formula, which often helps in understanding the general position of the curve on a graph.
  • Find additional points if necessary by picking x-values and calculating corresponding y-values, which supports the accuracy of your graph.
  • Ensure the direction of the parabola (up or down) is set correctly, indicating whether the function shows a maximum or minimum value at the vertex.
Graphing allows you to visualize how quadratic functions respond to changes in their coefficients.
Vertex
The vertex of a parabola is a critical point, either a maximum or minimum, and is located at \((h, k)\) in vertex form or \(\left(\frac{-b}{2a}, y\right)\) in standard form.
  • In the standard form \(ax^2 + bx + c\), use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate of the vertex.
  • Simply plug this x-value back into the equation to determine the y-coordinate, giving you the complete vertex point.
  • For our example, substituting into \( y = -3x^2 -5x + 3 \), we find the vertex at approximately \(\left(\frac{5}{6}, -1.42\right)\).
Finding and understanding the vertex allows you to fully interpret the graph鈥檚 highest or lowest point.

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