Parabola and Line Intersection Problem – Equations, Vertex, and Inequalities (1)

In this algebra problem, we explore the intersection of a parabola and a line using their graphs. Problems like this are common in Algebra and Pre-Calculus, since they combine concepts of quadratic functions, linear functions, and inequalities.

Problem:


The graphs of a parabola given by \( y = f(x) \) and that of a line given by \( y = g(x) \) are shown below. The parabola and the line intersect at points \( A \) and \( D \); where \( A \) is on the x-axis and has an x-coordinate equal to \( 2 \). Point \( V \) is the vertex of the parabola. parabola and line..
  1. Find the equation of \( g \) and write it of the form \( g(x) = m x + b \)
  2. Find the equation of the parabola.
  3. Find \( x \) coordinate of point \( D \).
  4. Use the graph to solve the inequality \( f(x) \gt g(x) \).

Solution to Problem 1:

  1. \( y = g(x) \) is the equation of a line through the points \( C(-3,-5) \) and \( A(2,0) \).

    slope is \[ (0-(-5)) / (2 -(-3)) = 1 \]

    equation of line: \[ y - 0 = 1(x - 2) \] , which may be written as: \[ y = x - 2 \]

    hence \[ g(x) = x - 2 \]

  2. The vertex of the parabola is known V(4,-4). The equation of the parabola in vertex form is given by \[ y = a(x - 4)^2 - 4 \] Point \( A(2,0) \) is on the parabola hence: \[ 0 = a(0 - 4)^2 - 4 \] solve the above equation for \( a \) to find \[ a = 1 \] hence the equation of the parabola is given by: \[ y = (x - 4)^2 - 4 \]
  3. The \( x \) coordinate of point \( D \) is determined by solving the equation \[ (x - 4)2 - 4 = x - 2 \] Solve the above equation to obtain the solutions \( x = 2 \) and \( x = 7 \).

    \( x = 2 \) is already known, it is the \( x \) coordinate of point \( A \).

    \( x = 7 \) is the \( x \) coordinate of point \( D \).

  4. From the left side of the graph to point A, f(x) > g(x) and therefore the interval \( (-\infty , 2) \) is a set of solutions. Also from point \( D \) to the right side of the graph \( f(x) \gt g(x) \) and therefore another set of solutions is given by the interval \( (7 , + \infty) \).

    The solution set of the inequality f(x) > g(x) is given by: \[ (- \infty , 2) U (7 , + \infty) \]

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