When you graph quadratic functions, there are some important features you should look at. These features help you see what the graph looks like and how it works.
Vertex: The vertex is the point where the parabola changes direction. You can find this point using the formula (x = -\frac{b}{2a}) if the function is written like (y = ax^2 + bx + c). This formula gives you the x-coordinate of the vertex. Then, you can put that x value back into the equation to find the y-coordinate. The vertex is important because it's where the highest or lowest value of the graph is found.
Axis of Symmetry: This is a line that goes straight up and down through the vertex. It shows that the graph is the same on both sides. You can find this line using (x = -\frac{b}{2a}) too, which is pretty handy!
Y-Intercept: The y-intercept is the spot where the graph hits the y-axis. To find it, just plug in (x = 0) into the function. This gives you the point ((0, c)) when you use the standard form (y = ax^2 + bx + c). This point helps give your graph a solid starting point.
X-Intercepts (Roots): These are the points where the graph crosses the x-axis, known as the roots or x-intercepts. You can find them with the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}). Knowing where these points are helps you understand how the graph behaves.
Direction of Opening: The number (a) in the standard form (y = ax^2 + bx + c) tells you if the parabola opens up or down. If (a) is greater than zero ((a > 0)), the graph opens up. If (a) is less than zero ((a < 0)), it opens down. This is key to seeing the overall shape of the graph.
By examining these features, you can create a clear and accurate picture of any quadratic function. Each detail helps you better understand how the function acts on the graph!
When you graph quadratic functions, there are some important features you should look at. These features help you see what the graph looks like and how it works.
Vertex: The vertex is the point where the parabola changes direction. You can find this point using the formula (x = -\frac{b}{2a}) if the function is written like (y = ax^2 + bx + c). This formula gives you the x-coordinate of the vertex. Then, you can put that x value back into the equation to find the y-coordinate. The vertex is important because it's where the highest or lowest value of the graph is found.
Axis of Symmetry: This is a line that goes straight up and down through the vertex. It shows that the graph is the same on both sides. You can find this line using (x = -\frac{b}{2a}) too, which is pretty handy!
Y-Intercept: The y-intercept is the spot where the graph hits the y-axis. To find it, just plug in (x = 0) into the function. This gives you the point ((0, c)) when you use the standard form (y = ax^2 + bx + c). This point helps give your graph a solid starting point.
X-Intercepts (Roots): These are the points where the graph crosses the x-axis, known as the roots or x-intercepts. You can find them with the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}). Knowing where these points are helps you understand how the graph behaves.
Direction of Opening: The number (a) in the standard form (y = ax^2 + bx + c) tells you if the parabola opens up or down. If (a) is greater than zero ((a > 0)), the graph opens up. If (a) is less than zero ((a < 0)), it opens down. This is key to seeing the overall shape of the graph.
By examining these features, you can create a clear and accurate picture of any quadratic function. Each detail helps you better understand how the function acts on the graph!