Learn on PengiCalifornia Reveal Math, Algebra 1Unit 2: Relations and Functions

2-5 Shapes of Graphs

In this Grade 9 lesson from California Reveal Math Algebra 1 (Unit 2: Relations and Functions), students learn to analyze the shapes of function graphs by identifying line symmetry and the line of symmetry, determining where a function is increasing or decreasing, and locating extrema including maximum, relative maximum, minimum, and relative minimum values. Students also explore end behavior of functions using real-world contexts such as suspension bridge cables and water fountain arcs. These concepts build foundational skills for interpreting and describing the behavior of functions across different intervals.

Section 1

Defining Line of Symmetry

Property

The axis of symmetry is a vertical line that divides a graph into two congruent, mirror-image halves. The equation of this vertical line is given by x=hx = h, where hh is a constant. For any point on the graph, there is a corresponding point on the opposite side of this line that is equidistant from it.

Examples

  • If a parabola has an axis of symmetry at x=2x = 2 and a point at (0,5)(0, 5), there must be a corresponding mirror-image point at (4,5)(4, 5).
  • An absolute value function with a vertex at (−3,1)(-3, 1) has an axis of symmetry with the equation x=−3x = -3.
  • If a graph has an axis of symmetry at x=0x=0 (the y-axis), then for any point (x,y)(x, y) on the graph, the point (−x,y)(-x, y) is also on the graph.

Explanation

The axis of symmetry is a fundamental property of certain functions, most notably quadratic and absolute value functions. It is a vertical line that passes through the vertex of the graph. Understanding the axis of symmetry allows you to predict the location of points on the graph, as every point (except those on the axis itself) has a matching counterpart on the other side.

Section 2

Finding the Line of Symmetry by Checking Equidistant Points

Property

A graph has a vertical line of symmetry at x=ax = a if, for every point (a+d, y)(a + d,\ y) on the graph, there is a corresponding point (a−d, y)(a - d,\ y) with the same yy-value — that is, points at equal horizontal distances dd on either side of x=ax = a share the same height.

Step-by-step procedure:

Section 3

Identifying Increasing and Decreasing Intervals

Property

A function is increasing on an interval if as xx values move from left to right, the yy values rise (positive slope).
A function is decreasing on an interval if as xx values move from left to right, the yy values fall (negative slope).

Examples

Section 4

Defining Absolute Maximum and Minimum Values

Property

A function has a maximum value when there is a yy-value that is greater than or equal to all other yy-values in the function''s range. A function has a minimum value when there is a yy-value that is less than or equal to all other yy-values in its range. These values are also known as global or absolute maximums and minimums.

Examples

  • The function f(x)=−x2+4f(x) = -x^2 + 4 has a maximum value of 44. It has no minimum value.
  • The function g(x)=∣x∣−2g(x) = |x| - 2 has a minimum value of −2-2. It has no maximum value.
  • The function h(x)=sin⁡(x)h(x) = \sin(x) has a maximum value of 11 and a minimum value of −1-1.

Explanation

The maximum and minimum are the "highest" and "lowest" points on the entire graph of a function. The maximum value is the largest output (yy-value) the function can produce, while the minimum value is the smallest output. Not all functions have a maximum or minimum value; for example, a line like y=xy=x continues infinitely in both positive and negative yy directions.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Defining Line of Symmetry

Property

The axis of symmetry is a vertical line that divides a graph into two congruent, mirror-image halves. The equation of this vertical line is given by x=hx = h, where hh is a constant. For any point on the graph, there is a corresponding point on the opposite side of this line that is equidistant from it.

Examples

  • If a parabola has an axis of symmetry at x=2x = 2 and a point at (0,5)(0, 5), there must be a corresponding mirror-image point at (4,5)(4, 5).
  • An absolute value function with a vertex at (−3,1)(-3, 1) has an axis of symmetry with the equation x=−3x = -3.
  • If a graph has an axis of symmetry at x=0x=0 (the y-axis), then for any point (x,y)(x, y) on the graph, the point (−x,y)(-x, y) is also on the graph.

Explanation

The axis of symmetry is a fundamental property of certain functions, most notably quadratic and absolute value functions. It is a vertical line that passes through the vertex of the graph. Understanding the axis of symmetry allows you to predict the location of points on the graph, as every point (except those on the axis itself) has a matching counterpart on the other side.

Section 2

Finding the Line of Symmetry by Checking Equidistant Points

Property

A graph has a vertical line of symmetry at x=ax = a if, for every point (a+d, y)(a + d,\ y) on the graph, there is a corresponding point (a−d, y)(a - d,\ y) with the same yy-value — that is, points at equal horizontal distances dd on either side of x=ax = a share the same height.

Step-by-step procedure:

Section 3

Identifying Increasing and Decreasing Intervals

Property

A function is increasing on an interval if as xx values move from left to right, the yy values rise (positive slope).
A function is decreasing on an interval if as xx values move from left to right, the yy values fall (negative slope).

Examples

Section 4

Defining Absolute Maximum and Minimum Values

Property

A function has a maximum value when there is a yy-value that is greater than or equal to all other yy-values in the function''s range. A function has a minimum value when there is a yy-value that is less than or equal to all other yy-values in its range. These values are also known as global or absolute maximums and minimums.

Examples

  • The function f(x)=−x2+4f(x) = -x^2 + 4 has a maximum value of 44. It has no minimum value.
  • The function g(x)=∣x∣−2g(x) = |x| - 2 has a minimum value of −2-2. It has no maximum value.
  • The function h(x)=sin⁡(x)h(x) = \sin(x) has a maximum value of 11 and a minimum value of −1-1.

Explanation

The maximum and minimum are the "highest" and "lowest" points on the entire graph of a function. The maximum value is the largest output (yy-value) the function can produce, while the minimum value is the smallest output. Not all functions have a maximum or minimum value; for example, a line like y=xy=x continues infinitely in both positive and negative yy directions.