Understanding Graphs of Equations: Sketching, Intercepts, and Symmetry
This guide covers essential concepts in graphing equations, including how to sketch graphs, find x- and y-intercepts, and utilize symmetry. Learn methods to graph circles and apply these techniques to real-life problems. By understanding the Cartesian plane and important examples, you will gain the skills needed to visualize mathematical functions effectively. This guide also introduces the distance formula and the standard equation of a circle, reinforcing your grasp of graphing fundamentals.
Understanding Graphs of Equations: Sketching, Intercepts, and Symmetry
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Presentation Transcript
Section 1.1 Graphs of Equations
What you should learn • How to sketch graphs of equations • How to find x- and y-intercepts of graphs of equations • How to use symmetry to sketch graphs of equations • How to find equations and sketch graphs of circles • How to use graphs of equations in solving real-life problems
Rectangular Coordinate System Origin y-axis Quadrants II I x-axis IV III Cartesian Plane
Graph y = -2x + 1 • (0,1) • ( -1, 3) • ( 1, -1)
Calculator • [y = ] • Enter y = 5 – 2x • [GRAPH] • [2ND] [TABLE] • Can you get this onto a piece of paper?
Calculator • [y = ] • Enter y = x^2 – 5 • [GRAPH] • [2ND] [TABLE] • Can you get this onto a piece of paper?
Intercepts of a Graph What can you tell me about intercepts?
Intercepts of a Graph What can you tell me about intercepts?
What can you tell me about intercepts? Intercepts of a Graph
Intercepts of a Graph What can you tell me about intercepts?
(0, y) (x, 0) Finding Intercepts • To find x-intercepts, let y be zero and solve the equation for x. • To find y-intercepts, let x be zero and solve the equation for y.
Example C: Find the x- and y-intercepts of : (x, 0) (0, y)
Example D: Find the x- and y-intercepts of : (x, 0) (0, y)
Calculator • [y = ] • Enter y = (x+4)^.5 • Or y = √(x+4) • [GRAPH] • [2ND] [TABLE] • Can you get this onto a piece of paper? (x, 0) (0, y)
Example E: Find the x- and y-intercepts of : (x, 0) (0, y)
Calculator • [y = ] • Enter y = abs(2x+5) • To get “abs” • [math] Select NUM select [1 abs] • [GRAPH] • [2ND] [TABLE] • Can you get this onto a piece of paper?
Example F: Find the x- and y-intercepts of : (x, 0) (0, y)
Calculator • [y = ] • Enter y = 2x^3– 4x^2 –6x • [GRAPH] • [2ND] [TABLE] • Can you get this onto a piece of paper?
3 Flavors of Symmetry • X – Axis Symmetry • Fold the x-axis • (x, y) (x, -y) • Y – Axis Symmetry • Fold the y-axis • (x, y) (-x, y) • Origin Symmetry • Spin • (x, y) (-x, -y)
Example B X – Axis SymmetryFold the x-axis (x, y) (x, -y)
Example B Y – Axis SymmetryFold the y-axis (x, y) (-x, y)
Algebraic Tests for Symmetry • The graph of an equation is symmetric to the x-axis if replacing y with –y yields an equivalent equation. Since this is the equation we started with we know that the relation has x-axis symmetry.
Algebraic Tests for Symmetry • The graph of an equation is symmetric to the y-axis if replacing x with –x yields an equivalent equation. Since this equation is different than what we started with we know that the relation does not have y-axis symmetry.
a b Given:-Square-Length a a c c b Are the triangles congruent? c b c a What type of quadrilateral is formed by connecting the points? b a What is the area of the large square? What is the area of the small square? What is the area of all four triangles? = +
a b Pythagorean Theorem (a+b)2 a c2 c c b c b c a b a What is the area of the large square? What is the area of the small square? What is the area of all four triangles? = +
Circle • What is the radius? • What is a circle?
Homework 1-31 odd, 57-71 odd 76
