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Magazine Chapter 5: Problem 6
Complete parts a-c for each quadratic function. a. Find the \(y\) -intercept, the equation of the axis of symmetry, and the \(x\) -coordinate of the vertex. b. Make a table of values that includes the vertex. c. Use this information to graph the function. $$ f(x)=3 x^{2}+10 x $$
Short Answer
Expert verified
y-intercept: 0; axis of symmetry: \(x = -\frac{5}{3}\); vertex: \((-\frac{5}{3}, -\frac{25}{3})\). Graph is a parabola opening upwards.
Step by step solution
01
Find the y-intercept
To find the y-intercept of the quadratic function, substitute \(x = 0\) into the equation \(f(x) = 3x^2 + 10x\). This gives us \(f(0) = 3(0)^2 + 10(0) = 0\). Thus, the y-intercept is \(0\).
02
Determine the Equation of the Axis of Symmetry
The equation for the axis of symmetry of a quadratic function \(ax^2 + bx + c\) is given by \(x = -\frac{b}{2a}\). For the given function, \(a = 3\) and \(b = 10\). Substituting, \(x = -\frac{10}{2 imes 3} = -\frac{10}{6} = -\frac{5}{3}\). Thus, the axis of symmetry is \(x = -\frac{5}{3}\).
03
Find the x-coordinate of the Vertex
The x-coordinate of the vertex is the same as the axis of symmetry: \(x = -\frac{5}{3}\).
04
Calculate the y-coordinate of the Vertex
Substitute the x-coordinate of the vertex \(-\frac{5}{3}\) into the function to find the y-coordinate: \[f\left(-\frac{5}{3}\right) = 3\left(-\frac{5}{3}\right)^2 + 10\left(-\frac{5}{3}\right) = 3\times\frac{25}{9} - \frac{50}{3}\] Solving this gives:\[= \frac{75}{9} - \frac{150}{9} = \frac{-75}{9} = -\frac{25}{3}\] Thus, the vertex is \((-\frac{5}{3}, -\frac{25}{3})\).
05
Create a Table of Values
Choose values of \(x\) around the vertex \(x = -\frac{5}{3}\). Calculate the corresponding \(f(x)\) for \(x = -2\), \(-1\), \(0\), and \(-3\):- \(f(-3) = 3(-3)^2 + 10(-3) = 27 - 30 = -3\)- \(f(-2) = 3(-2)^2 + 10(-2) = 12 - 20 = -8\)- \(f(-1) = 3(-1)^2 + 10(-1) = 3 - 10 = -7\)- \(f(0) = 0\)| \(x\) | \(f(x)\) ||-------|---------|| -3 | -3 || -2 | -8 || -1 | -7 || 0 | 0 || -5/3 | -25/3 |
06
Graph the Function
To graph the function, plot the points from the table of values and the vertex \((-\frac{5}{3}, -\frac{25}{3})\). Draw the axis of symmetry at \(x = -\frac{5}{3}\). Sketch the parabola opening upwards as the leading coefficient \(3 > 0\). Make sure it fits through all points, maintaining symmetry relative to the line \(x = -\frac{5}{3}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Y-Intercept
The y-intercept of a quadratic function is where the graph intersects the y-axis. This point occurs when the x-value is zero. To find the y-intercept, substitute \(x = 0\) into the quadratic equation. For the function \(f(x) = 3x^2 + 10x\), it becomes:
- \(f(0) = 3(0)^2 + 10(0) = 0\)
Axis of Symmetry
In the graph of a quadratic function, the axis of symmetry is a vertical line that passes through the vertex. It divides the parabola into two mirror-image halves. The equation for the axis of symmetry can be calculated using the formula:\[x = -\frac{b}{2a}\]where \(a\) and \(b\) are coefficients from the function \(ax^2 + bx + c\). For our function, \(a = 3\) and \(b = 10\), resulting in:
- \(x = -\frac{10}{6} = -\frac{5}{3}\)
Vertex
The vertex of a parabola is its highest or lowest point, depending on whether it opens upwards or downwards. For quadratic functions opening upwards, like \(f(x) = 3x^2 + 10x\), the vertex is the minimum point. The x-coordinate of the vertex is the same as the axis of symmetry, \(-\frac{5}{3}\). To find the y-coordinate, plug this x-value into the function:\[f\left(-\frac{5}{3}\right) = 3\left(-\frac{5}{3}\right)^2 + 10\left(-\frac{5}{3}\right) = -\frac{25}{3}\]Thus, the vertex of the graph is at \((-\frac{5}{3}, -\frac{25}{3})\). Knowing the vertex is key because it represents the tip of the parabola, giving insight into the graph's shape and direction.
Parabola
A parabola is the shape of the graph of a quadratic function. It has a curved, symmetrical arc that can open either upwards or downwards:
- If the leading coefficient (in front of \(x^2\)) is positive, like 3 in \(f(x) = 3x^2 + 10x\), the parabola opens upwards.
- If it is negative, the parabola opens downwards.
Graphing Quadratics
Graphing quadratic functions involves plotting the y-intercept, vertex, and additional points to sketch the parabola. Begin by identifying the intercept, axis of symmetry, and vertex coordinates to set key points. Use a table of values to determine other points through which the graph should pass. For example:
- Use points such as \((-3, -3)\), \((-2, -8)\), \((-1, -7)\), and \((0, 0)\) as determined in the calculation process.
- Draw the axis of symmetry (e.g., \(x = -\frac{5}{3}\)) as a dashed line for reference.
- Sketch the parabola, ensuring symmetry about this axis and that the graph passes through all plotted points.
