2.4.3 Solving equations graphically

• If the equation of a linear function is y = 2x - 3, what is the slope and y-intercept?

  Slope = 2, y-intercept = -3
• What is the general equation for a linear function?

  The general equation for a linear function is:

  y = mx + b

  where m is the slope and b is the y-intercept.
• Steps to solve equations graphically

  1️⃣ Plot the first equation as a graph

  2️⃣ Plot the second equation on the same graph

  3️⃣ Identify the points where the graphs intersect

  4️⃣ Read the coordinates of the intersection points
• Match the function type with its key feature:

  Linear ↔️ Straight line

  Quadratic ↔️ Parabola with vertex

  Rational ↔️ Asymptotes

  Logarithmic ↔️ No negative values
• What does solving equations graphically involve?

  Finding graph intersections
• What is the equation form of a linear function?

  $y =$$mx +$$c$
• What is the general equation for a rational function?

  The general equation for a rational function is:

  y = (x + 3) / (x - 2)
• What is the key feature of a quadratic function?

  Vertex
• What is the primary goal when determining the x-coordinate of the intersection point?

  Find the value of x
• What is the point of intersection of two graphs called?

  Intersection point
• If the equation of a quadratic function is y = (1/2)x^2 - 4, what are the coefficients a, b, and c?

  a = 1/2, b = 0, c = -4
• According to the graph, what is the y-intercept of the line $x - 2y +$$8 =$$0$?


  (0, 4)
• What is the x-coordinate of the intersection point for the equations $y =$$x +$$1$ and $y =$$- x +$$3$?


  1
• What are the four types of functions shown in the image?

  Linear, quadratic, rational, and logarithmic
• Equations with three variables cannot be solved graphically.

  False
• What is the general equation for a logarithmic function?

  The general equation for a logarithmic function is:

  y = log x
• If the equation of a logarithmic function is y = log x, what is the base of the logarithm?

  Base 10
• If the equation of a rational function is y = (x + 3) / (x - 2), what are the x-intercepts?

  x = -3, x = 2
• What type of equations are typically solved graphically?

  Equations with two variables
• What is the general equation for a quadratic function?

  The general equation for a quadratic function is:

  y = ax^2 + bx + c

  where a, b, and c are constants.
• What is the equation of the second line shown on the graph?

  $x - 2y +$$8 =$$0$
• Which point lies on the line defined by the equation $x - 2y +$$8 =$$0$?


  (4, 6)
• What is the equation of the first line shown on the graph?

  $2x - y +$$1 =$$0$
• How many points of intersection can a line and a curve have?

  Zero, one, or two
• What does the intersection point (2, 5) on the graph represent in relation to the two lines?

  The solution to both equations
• What happens to the value of $y$ as $x$ increases in the equation $2x - y +$$1 =$$0$?


  $y$ also increases
• What is the behavior of a logarithmic function?

  Increases slowly
• Which point lies on the line defined by the equation $2x - y +$$1 =$$0$?


  (3, 7)
• Based on the graph, what is the solution to the system of equations represented by the two lines?

  (2, 5)
• What is the graphical interpretation of a system of two linear equations?

  • Each linear equation represents a straight line on the coordinate plane.
  • The solution to the system is the point where the two lines intersect.
  • If the lines are parallel, there is no solution (the system is inconsistent).
  • If the lines are the same, there are infinitely many solutions (the system is dependent).
• According to the graph, what is the y-intercept of the line $2x - y +$$1 =$$0$?


  (0, 1)
• What happens to the value of $y$ as $x$ increases in the equation $x - 2y +$$8 =$$0$?


  $y$ also increases