I want to teach my students about the quadratic graphs. How different forms of quadratic equations tell us about the graph

视频信息

答案文本 复制

视频字幕 复制

Welcome to our lesson on quadratic functions! Today we'll explore how different forms of quadratic equations reveal specific information about parabolas. There are three main forms: standard form, vertex form, and factored form. Each form makes certain features of the graph immediately visible. Let's start with standard form: y equals a x squared plus b x plus c. The coefficient 'a' tells us if the parabola opens up or down, and how wide it is. The constant 'c' gives us the y-intercept directly. To find the vertex, we use negative b divided by two a for the x-coordinate. Let's see this with our example: y equals two x squared plus four x minus six. Now let's explore vertex form: y equals a times x minus h squared plus k. This form is incredibly useful because it shows the vertex coordinates directly as h and k. The vertex is at the point h comma k, and the axis of symmetry is the line x equals h. In our example, y equals negative three times x minus one squared plus five, we can immediately see the vertex is at one comma five. Finally, let's examine factored form: y equals a times x minus r one times x minus r two. This form immediately reveals the x-intercepts or roots of the parabola. The values r one and r two are where the graph crosses the x-axis. The vertex lies exactly halfway between these roots. In our example, y equals zero point five times x minus four times x plus two, we can see the roots are at four and negative two, so the vertex x-coordinate is their average: one. To summarize what we have learned: Standard form shows the y-intercept directly and provides a formula for the vertex. Vertex form makes the vertex coordinates immediately visible. Factored form reveals the x-intercepts at a glance. Each form highlights different features of the parabola, so choose the form that best shows what you need to analyze.