How to Graph Quadratic Equations in Standard Form

Quadratic equations are an interesting part of algebra that you'll learn about in Grade 10. The standard form of a quadratic equation looks like this:

$$y = ax^2 + bx + c$$

Here's what the letters mean:

• $y$ is the output or result,
• $x$ is the input or what you put into the equation,
• $a$, $b$, and $c$ are numbers, and $a$ cannot be zero.

In this equation, the $a$ value shows if the curve goes up or down and how wide it is. The $b$ value helps to find the vertex (the highest or lowest point of the curve), and $c$ helps to pinpoint where the curve crosses the y-axis. Knowing how to graph quadratic equations in standard form helps you see how they relate to each other.

Graphing Quadratic Equations

When you graph a quadratic equation, you're drawing a curve called a parabola. Depending on what $a$ is, this curve can open either up or down.

1. Opens Up: If $a > 0$, the parabola opens upwards and has a lowest point (the vertex).
2. Opens Down: If $a < 0$, it opens downwards and has the highest point.

Let's look at an example:

$$y = 2x^2 + 3x - 5$$

In this equation, $a = 2$, $b = 3$, and $c = -5$. Since $a > 0$, we know our parabola opens upwards.

Finding the Vertex

To graph this correctly, we need to find the vertex using this formula:

$$x = -\frac{b}{2a}$$

Now, let’s plug in our numbers:

$$x = -\frac{3}{2(2)} = -\frac{3}{4}$$

Next, we substitute $x = -\frac{3}{4}$ back into the equation to find $y$:

$$y = 2\left(-\frac{3}{4}\right)^2 + 3\left(-\frac{3}{4}\right) - 5$$

When we calculate that, we get:

$$y = 2\left(\frac{9}{16}\right) - \frac{9}{4} - 5 = \frac{9}{8} - \frac{18}{8} - \frac{40}{8} = -\frac{49}{8}$$

So the vertex is at the point $\left(-\frac{3}{4}, -\frac{49}{8}\right)$.

Finding the y-intercept

Next, we find where the curve crosses the y-axis. This happens when $x = 0$:

$$y = c = -5$$

So the y-intercept is the point $(0, -5)$.

Sketching the Graph

Now that we have the vertex at $\left(-\frac{3}{4}, -\frac{49}{8}\right)$ and the y-intercept at $(0, -5)$, we can draw the graph:

1. Start by plotting the vertex.
2. Then plot the y-intercept.
3. Draw a smooth, U-shaped curve (the parabola) around the vertex.

You can also pick a few other $x$ values to find more points on the curve. This will help you create a clearer graph.

Remember, the more points you have, the better your graph will show what the quadratic function looks like! Learning to graph quadratic equations isn't just about drawing lines. It’s about understanding how each part of the equation works together to form that beautiful curve. Have fun graphing!