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Magazine Chapter 3: Problem 21
Graph the given functions. $$y=x^{2}-3 x+1$$
Short Answer
Expert verified
The graph is a parabola with vertex (\(\frac{3}{2}, -\frac{5}{4}\)) and y-intercept (0, 1), opening upwards.
Step by step solution
01
Identify the Function Type
The given function is a quadratic function, which can be identified by its form, \(y = ax^2 + bx + c\). Here, \(a = 1\), \(b = -3\), and \(c = 1\). The graph of a quadratic function is a parabola.
02
Find the Vertex of the Parabola
The vertex of a parabola given by the equation \(y = ax^2 + bx + c\) can be found using the formula \(x = \frac{-b}{2a}\). For this function, \(x = \frac{-(-3)}{2(1)} = \frac{3}{2}\). Substitute \(x = \frac{3}{2}\) back into the function to find \(y\): \(y = (\frac{3}{2})^2 - 3(\frac{3}{2}) + 1\). This simplifies to \(y = \frac{9}{4} - \frac{9}{2} + 1 = -\frac{5}{4}\). So, the vertex is \((\frac{3}{2}, -\frac{5}{4})\).
03
Determine the Axis of Symmetry
The axis of symmetry can be found at \(x = \frac{3}{2}\), which is the x-value of the vertex. This is a vertical line that splits the parabola into two symmetrical halves.
04
Identify the Y-intercept
The y-intercept is the point where the graph crosses the y-axis, which occurs when \(x = 0\). Substitute \(x = 0\) into the function: \(y = 0^2 - 3(0) + 1 = 1\). Thus, the y-intercept is \((0, 1)\).
05
Plot the Graph
Start by plotting the vertex \((\frac{3}{2}, -\frac{5}{4})\) and the y-intercept \((0, 1)\). Use the axis of symmetry as a guide. Since \(a = 1\) is positive, the parabola opens upwards. Plot additional points on either side of the axis of symmetry if needed, and draw a smooth, U-shaped curve through these points. The graph represents the quadratic function \(y = x^2 - 3x + 1\).
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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 follows the standard form \(y = ax^2 + bx + c\). These are characterized by the squared term \(x^2\), which makes their graphs take the shape of a parabola. In our specific example, the quadratic function is given by \(y = x^2 - 3x + 1\). Here, the coefficients are \(a = 1\), \(b = -3\), and \(c = 1\).
- The term \(a\) determines the "width" and the direction of the parabola (open upwards if positive, downwards if negative).
- The term \(b\) influences the position of the vertex along the x-axis.
- The term \(c\) is the constant and dictates where the parabola crosses the y-axis (known as the y-intercept).
Vertex of a Parabola
The vertex of a parabola is its highest or lowest point, depending on whether the parabola opens upwards or downwards. For a quadratic in standard form \(y = ax^2 + bx + c\), the vertex can be found using the formula \(x = \frac{-b}{2a}\).
Substitute the values from the function into this formula, where \(b = -3\) and \(a = 1\), to get \(x = \frac{3}{2}\).
To find the y-coordinate of the vertex, replace \(x\) in the original equation: \[y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) + 1\] Simplifying, gives \(y = -\frac{5}{4}\).
Substitute the values from the function into this formula, where \(b = -3\) and \(a = 1\), to get \(x = \frac{3}{2}\).
To find the y-coordinate of the vertex, replace \(x\) in the original equation: \[y = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) + 1\] Simplifying, gives \(y = -\frac{5}{4}\).
- The vertex for \(y = x^2 - 3x + 1\) is at \(\left(\frac{3}{2}, -\frac{5}{4}\right)\).
Axis of Symmetry
The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. It passes through the vertex. For the quadratic function \(y = x^2 - 3x + 1\), the x-coordinate of the vertex, \(\frac{3}{2}\), gives the equation for the axis of symmetry:\[ x = \frac{3}{2} \]
- This line helps in graphing the parabola accurately, as any point on one side has a corresponding point on the opposite side of the axis.
- The axis of symmetry is especially useful since it makes it easier to verify that your graph is correctly plotted and symmetrical.
Y-intercept
The y-intercept of a graph is the point where it crosses the y-axis. For any function, this occurs where x equals zero. For our quadratic function \(y = x^2 - 3x + 1\), substitute \(x = 0\):\[y = 0^2 - 3(0) + 1 = 1\]
- This means the y-intercept is at \((0, 1)\).
- Knowing the y-intercept allows you to begin plotting the graph since it offers a reliable point that the curve must pass through.
