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Magazine Chapter 0: Problem 18
Graph. $$ y=x^{2}-4 x+3 $$
Short Answer
Expert verified
Graph is a parabola, opening upwards, vertex at (2,1), crossing the x-axis at (1,0) and (3,0), and the y-axis at (0,3).
Step by step solution
01
Identify the type of equation
The equation given is \(y = x^2 - 4x + 3\). This is a quadratic equation, which is typically represented by a parabolic curve on a graph.
02
Determine the vertex
The vertex form of a quadratic equation is \(y = a(x-h)^2 + k\), where \((h,k)\) is the vertex. To find \(h\), use the formula \(h = -\frac{b}{2a}\) for the equation \(ax^2 + bx + c\). Here, \(a = 1\), \(b = -4\), so \(h = -\frac{-4}{2 \cdot 1} = 2\). Substitute \(x = 2\) back into the equation to find \(k\): \(y = (2)^2 - 4(2) + 3 = 1\). Thus, the vertex is \((2, 1)\).
03
Identify the axis of symmetry
The axis of symmetry for a parabola given by \(ax^2 + bx + c\) is the vertical line \(x = h\). From the previous step, \(h = 2\), so the axis of symmetry is \(x = 2\).
04
Calculate y-intercept
To find the y-intercept, set \(x = 0\) in the equation, which gives \(y = 0^2 - 4(0) + 3 = 3\). Thus, the y-intercept is \( (0, 3) \).
05
Find x-intercepts (roots)
To find the x-intercepts, set \(y = 0\) and solve \(x^2 - 4x + 3 = 0\). Factor the equation: \((x-3)(x-1) = 0\), giving roots \(x = 3\) and \(x = 1\). So, the x-intercepts are \((3,0)\) and \((1,0)\).
06
Plot and draw the graph
With the vertex \((2,1)\), axis of symmetry \(x = 2\), y-intercept \((0,3)\), and x-intercepts \((3,0)\) and \((1,0)\), plot these points and draw a smooth parabolic curve opening upwards.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabolic Curves
A parabolic curve, often simply called a parabola, is a symmetric, u-shaped curve. Quadratic equations like \(y = x^2 - 4x + 3\) create these parabolic shapes on a graph. Parabolas have a distinct and recognizable form due to their symmetry and the way they open either upwards or downwards.
- The direction of the parabola depends on the coefficient \(a\) in the general quadratic equation \(ax^2 + bx + c\). If \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.
- Symmetry is a key feature; each side of the vertex mirrors the other, which is very helpful in graphing and understanding their behavior on a coordinate plane.
Vertex of a Parabola
The vertex is a crucial point on the parabola and is essentially the turning point. It's where the curve changes direction. For the equation \(y = x^2 - 4x + 3\), the vertex is calculated using the formula for the vertex, \((h, k)\), derived from the vertex form \(y = a(x-h)^2 + k\).
To find \(h\), the x-coordinate of the vertex, use \(h = -\frac{b}{2a}\). With \(a = 1\) and \(b = -4\), this gives \(h = 2\).
Substitute \(x = 2\) back into the original quadratic equation to find \(k\), giving \(y = 1\). Thus, the vertex is \((2, 1)\).
To find \(h\), the x-coordinate of the vertex, use \(h = -\frac{b}{2a}\). With \(a = 1\) and \(b = -4\), this gives \(h = 2\).
Substitute \(x = 2\) back into the original quadratic equation to find \(k\), giving \(y = 1\). Thus, the vertex is \((2, 1)\).
- The vertex not only marks the curve's highest or lowest point, but it also sits precisely on the axis of symmetry, providing a constant reference point to measure the parabola's symmetry around itself.
- In applied contexts, knowing the vertex can be important for maximization or minimization problems.
X-intercepts
X-intercepts occur where the parabola crosses the x-axis. These intersections are vital as they often represent the solution or the roots of the quadratic equation. To find these intercepts for \(y = x^2 - 4x + 3\), set \(y = 0\) and solve \(x^2 - 4x + 3 = 0\).
- Factoring the equation gives \((x-3)(x-1) = 0\), so the roots or the x-intercepts are \(x = 3\) and \(x = 1\).
- These points are seen on the graph as \((3,0)\) and \((1,0)\), where the curve meets the x-axis.
- The x-intercepts can be visualized as the points where the baseball hits the ground or where an archway touches the base.
Y-intercepts
Y-intercepts show where the parabolic curve intersects the y-axis, revealing vital information about the function's value when \(x = 0\). For the equation \(y = x^2 - 4x + 3\), the y-intercept can be determined by simply evaluating the equation where \(x = 0\).
- Setting \(x = 0\), we find \(y = 3\), hence the point \((0, 3)\) is our y-intercept.
- The y-intercept provides insight into the graph's starting point on the y-axis, giving a base for sketching the parabola accurately.
- This intercept often represents an initial condition or starting position in various applications.
