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2.2: Properties of Quadratic Functions

Learning Objectives

  1. Find key points on the graph of a quadratic function, including the vertex, intercepts, and minimum/maximum value.
  2. Graph a quadratic function.

Graph a quadratic function

A quadratic equation has

Standard form [latex]ax^2+bx+c=0[/latex], where [latex]a, b,[/latex] and [latex]c[/latex] are real numbers and [latex]a\neq0[/latex].

Vertex form [latex]a\left(x-h\right)^2+k=0[/latex], where [latex]a[/latex] is real numbers and [latex]a\neq0[/latex].

The graph of a quadratic function is a U-shaped curve called a parabola. One important feature of the graph is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value. In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry.

A graph of a quadratic function, a parabola, is shown on a Cartesian coordinate system. The x-axis ranges from -6 to 6, and the y-axis ranges from -6 to 6, with grid lines at integer values. The parabola is U-shaped and opens upwards. Its vertex, the lowest point on the graph, is located at (1, -4). A vertical dashed red line passes through the vertex at x = 1, labeled 'Axis of symmetry'. The parabola intersects the x-axis at two points, (-1, 0) and (3, 0), which are labeled 'x-intercepts'. The parabola intersects the y-axis at the point (0, -3), which is labeled 'y-intercept'. Several points are plotted on the parabola, including (-2, 5) and (4, 5), illustrating its symmetric nature around the axis of symmetry. The label 'Vertex' points to the coordinate (1, -4).

Example 2.2-1-1: Find information on the graph.

  1. Is the quadratic a value positive or negative?
  2. What are the x-intercepts?
  3. What is the y-intercept?
  4. What is the axis of symmetry?
  5. Where is the vertex?
  6. Does it have a minimum or maximum? What is the value?
 A graph of a parabola is shown on a Cartesian coordinate system. The x-axis ranges from -10 to 10, and the y-axis ranges from -10 to 10, with grid lines at integer values. A red parabola opens upwards with its vertex located at (-2, -1). The curve is symmetric about a vertical line passing through x = -2. The parabola intersects the y-axis at approximately (0, 3) and intersects the x-axis at two points, approximately (-3, 0) and (-1, 0).

Master L3 Key

Example 2.2-1-1: Find information on the graph.

  1. Is the quadratic a value positive or negative?

a is positive, since the graph opens up.

  1. What are the x-intercepts?

(-3,0) and (-1,0)

x-intercept means cross or touch the x-axis, thus y must be 0.

 A graph of the quadratic function y = x squared + 4x + 3 is shown on a Cartesian coordinate system. The x-axis ranges from -6 to 6, and the y-axis ranges from -4 to 9, with grid lines at integer values. A blue parabola opens upwards. The vertex, the lowest point on the graph, is located at (-2, -1). A vertical dashed red line passes through the vertex at x = -2, labeled 'Axis of symmetry'. The parabola intersects the x-axis at two points, (-3, 0) and (-1, 0), which are labeled 'x-intercepts'. The parabola intersects the y-axis at the point (0, 3) labeled 'y-intercepts'. The equation of the quadratic function, 'y = x squared + 4x + 3', is written in blue in the upper right corner of the graph.
  1. What is the y-intercept?

(0,3)

y-intercept means cross or touch the y-axis, thus x must be 0.

  1. What is the axis of symmetry?

X= – 2

It can be found at the extreme point. It must be an equation, since it is a vertical line.

  1. Where is the vertex?

(-2,-1)

Extreme point on the graph.

  1. Does it have a minimum or maximum? What is the value?

It has a minimum, and it is -1.

The minimum /maximum value is the y value at the vertex.

Your Turn

Practice 2.2-1-1

Graph a quadratic function

Steps to Graph a Quadratic Function

  1. Find the x-intercepts (if possible):
    Set the quadratic equation equal to zero and solve for [latex]x[/latex]. These are the points where the graph crosses the x-axis.
  2. Find the y-intercept:
    Substitute [latex]x=0[/latex] into the equation to find the y-intercept. This is where the graph crosses the y-axis.
  3. Find the vertex:
    Use the formula [latex]x=\frac{-b}{2a}[/latex] to find the x-coordinate of the vertex. Then substitute this value into the equation to find the corresponding y-coordinate.
  4. Plot the key points:
    Plot the x-intercepts, y-intercept, and the vertex on the coordinate plane.
  5. Identify the axis of symmetry:
    Draw a vertical dashed line through the vertex. This line is the axis of symmetry and helps ensure the graph is balanced.
  6. If needed, find additional points:
    If you cannot find the x- or y-intercepts, or if you need more detail for accuracy, choose one or two additional x-values that haven’t been used, substitute them into the equation, and plot those points.
  7. Sketch the parabola:
    Connect all the plotted points smoothly in a U-shaped curve. Extend the curve on both sides, following the general shape and direction of a parabola (opening upward or downward based on the sign of [latex]a[/latex]).

Example 2.2-2-1: Graph

[latex]f\left(x\right)=x^2-2x-15[/latex]

Master L3 Key

Example 2.2-2-1: Graph

[latex]f\left(x\right)=x^2-2x-15[/latex]

  1. Find the x-intercepts (if possible):
    Set the quadratic equation equal to zero and solve for [latex]x[/latex]. These are the points where the graph crosses the x-axis.

[latex]x^2-2x-15=0[/latex]

An image illustrating to find two numbers that multiply to 'ac' and add up to 'b' for a quadratic function. The image is divided into two main sections. On the left, a large 'X' diagram is shown. The top of the 'X' is labeled 'ac = 15', representing the product of the 'a' and 'c' coefficients. The bottom of the 'X' is labeled 'b = -2', representing the 'b' coefficient. The two diagonal arms of the 'X' are used to find two numbers that multiply to 15 and add up to -2. These numbers are found to be 3 and -5, which are written in red text on the left and right arms of the 'X' respectively, labeled '#1' and '#2'. On the right, a table-like structure further explains the process. The top part reiterates 'ac = 15' and shows '|ac| = 15'. Below this, a green line separates the 'ac' value from two columns labeled '#1' and '#2', listing pairs of factors of 15. The pairs shown are (2, 15) and (3, -5). The pair '3' and '-5' is encircled in black, highlighting them as the solution because their product is 15 and their sum is -2.

[latex]\left(x+3\right)\left(x-5\right)=0[/latex]

[latex]x+3=0[/latex]

[latex]x_1=-3[/latex]

[latex]x-5=0[/latex]

[latex]x_2=5[/latex]

x-intercepts [latex]\left(-3,\;0\right),\;\left(5,\;0\right)[/latex]

  1. Find the y-intercept:
    Substitute [latex]x=0[/latex] into the equation to find the y-intercept. This is where the graph crosses the y-axis.

[latex]f\left(0\right)=\left(0\right)^2-2\left(0\right)-15=-15[/latex]

y-intercept [latex]\left(0,\;-15\right)[/latex]

  1. Find the vertex:
    Use the formula [latex]x=\frac{-b}{2a}[/latex] to find the x-coordinate of the vertex. Then substitute this value into the equation to find the corresponding y-coordinate.

[latex]x=\frac{-\left(-2\right)}{2\left(1\right)}=1[/latex]

[latex]y=f\left(1\right)=\left(1\right)^2-2\left(1\right)-15=-16[/latex]

Vertex [latex]\left(1,\;-16\right)[/latex]

  1. Plot the key points:
    Plot the x-intercepts, y-intercept, and the vertex on the coordinate plane.

A Cartesian coordinate plane is shown with the x-axis ranging from -10 to 10 and the y-axis ranging from -16 to 4, with grid lines at integer values. Four points are plotted on the plane and labeled. A purple point is located on the x-axis at (-3, 0) and labeled '(-3, 0)'. A black point is located on the x-axis at (5, 0) and labeled '(5, 0)'. These two points represent the x-intercepts. A red point is located on the y-axis at (0, -15) and labeled '(0, -15)', representing the y-intercept. A blue point is located at (1, -16) and labeled '(1, -16)'.

  1. Identify the axis of symmetry:
    Draw a vertical dashed line through the vertex. This line is the axis of symmetry and helps ensure the graph is balanced.

[latex]x=1[/latex]

  1. If needed, find additional points:
    If you cannot find the x- or y-intercepts, or if you need more detail for accuracy, choose one or two additional x-values that haven’t been used, substitute them into the equation, and plot those points.

We have all the points:

x-intercepts [latex]\left(-3,\;0\right),\;\left(5,\;0\right)[/latex]

y-intercept [latex]\left(0,\;-15\right)[/latex]

Vertex [latex]\left(1,\;-16\right)[/latex]

  1. Sketch the parabola:
    Connect all the plotted points smoothly in a U-shaped curve. Extend the curve on both sides, following the general shape and direction of a parabola (opening upward or downward based on the sign of [latex]a[/latex]).

A Cartesian coordinate plane is shown with the x-axis ranging from -10 to 10 and the y-axis ranging from -16 to 4, with grid lines at integer values. A red parabola opens upwards, passing through four labeled points. The parabola intersects the x-axis at two points: a purple point at (-3, 0) labeled '(-3, 0)' and a black point at (5, 0) labeled '(5, 0)', representing the x-intercepts. The parabola intersects the y-axis at a red point at (0, -15) labeled '(0, -15)', representing the y-intercept. The vertex, the lowest point of the parabola, is a blue point located at (1, -16) labeled '(1, -16)'.

Your Turn

Practice 2.2-2-1

  • Read the question carefully. Solve it using paper and pen.
  • When you are ready, select the answer arrow to reveal the correct answer and check your work.
  • If your answer is incorrect, try solving it again.
  • This process will help you practice and improve your skills.

Graph [latex]f\left(x\right)=x^2+8x+15[/latex]

Answer

A graph of a parabola opening upwards is shown on a Cartesian coordinate system. The x-axis ranges from -10 to 5, and the y-axis ranges from -5 to 15, with grid lines at integer values. The parabola is red and passes through several labeled points. Two purple points on the x-axis represent the x-intercepts: (-5, 0) and (-3, 0). A red point on the y-axis represents the y-intercept: (0, 15). The vertex, the lowest point of the parabola, is a blue point located at (-4, -1).

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