{"id":9440,"date":"2023-10-20T19:22:31","date_gmt":"2023-10-20T19:22:31","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=9440"},"modified":"2025-08-28T04:40:58","modified_gmt":"2025-08-28T04:40:58","slug":"quadratic-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/quadratic-functions-learn-it-3\/","title":{"raw":"Quadratic Functions: Learn It 3","rendered":"Quadratic Functions: Learn It 3"},"content":{"raw":"

Key Features of a Parabola's Graph<\/h2>\r\n

The graph of a quadratic function is a U-shaped curve called a\u00a0parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the\u00a0vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the\u00a0minimum value<\/strong>\u00a0of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the\u00a0maximum value<\/strong>. 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\u00a0axis of symmetry<\/strong>.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 1. A graph of a parabola with the x and y intercepts, vertex, and axis of symmetry labeled[\/caption]\r\n<\/center>
<\/center>\r\n

 <\/p>\r\n

The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis. The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the zeros<\/strong>, or roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex]\u00a0at which [latex]y=0[\/latex].<\/p>\r\n

\r\n

The places where a function's graph crosses the horizontal axis are the places where the function value equals zero. You've seen that these values are called\u00a0horizontal intercepts<\/em>, [latex]x[\/latex]-intercepts<\/em>,\u00a0and\u00a0zeros<\/em> so far. They can also be referred to as the\u00a0roots<\/em> of a function.<\/p>\r\n<\/section>\r\n

\r\n

Determine the vertex, axis of symmetry, zeros, and [latex]y[\/latex]-intercept of the parabola shown below.<\/p>\r\n

\"Graph<\/center>\r\n

[reveal-answer q=\"366804\"]Show Solution[\/reveal-answer]<\/p>\r\n

[hidden-answer a=\"366804\"]<\/p>\r\n

The vertex is the turning point of the graph. We can see that the vertex is at [latex](3,1)[\/latex]. The axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is [latex]x=3[\/latex]. This parabola does not cross the [latex]x[\/latex]-axis, so it has no zeros. It crosses the [latex]y[\/latex]-axis at [latex](0, 7)[\/latex] so this is the [latex]y[\/latex]-intercept.<\/p>\r\n

[\/hidden-answer]<\/p>\r\n<\/section>\r\n

[ohm2_question hide_question_numbers=1]13808[\/ohm2_question]<\/section>","rendered":"

Key Features of a Parabola’s Graph<\/h2>\n

The graph of a quadratic function is a U-shaped curve called a\u00a0parabola<\/strong>. One important feature of the graph is that it has an extreme point, called the\u00a0vertex<\/strong>. If the parabola opens up, the vertex represents the lowest point on the graph, or the\u00a0minimum value<\/strong>\u00a0of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the\u00a0maximum value<\/strong>. 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\u00a0axis of symmetry<\/strong>.<\/p>\n

\n
\"Graph
Figure 1. A graph of a parabola with the x and y intercepts, vertex, and axis of symmetry labeled<\/figcaption><\/figure>\n<\/div>\n
<\/div>\n

 <\/p>\n

The [latex]y[\/latex]-intercept is the point at which the parabola crosses the [latex]y[\/latex]-axis. The [latex]x[\/latex]-intercepts are the points at which the parabola crosses the [latex]x[\/latex]-axis. If they exist, the [latex]x[\/latex]-intercepts represent the zeros<\/strong>, or roots<\/strong>, of the quadratic function, the values of [latex]x[\/latex]\u00a0at which [latex]y=0[\/latex].<\/p>\n

\n

The places where a function’s graph crosses the horizontal axis are the places where the function value equals zero. You’ve seen that these values are called\u00a0horizontal intercepts<\/em>, [latex]x[\/latex]-intercepts<\/em>,\u00a0and\u00a0zeros<\/em> so far. They can also be referred to as the\u00a0roots<\/em> of a function.<\/p>\n<\/section>\n

\n

Determine the vertex, axis of symmetry, zeros, and [latex]y[\/latex]-intercept of the parabola shown below.<\/p>\n

\"Graph<\/div>\n