Explain the difference between a linear and a quadratic function

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Let's explore the difference between linear and quadratic functions. Linear functions have the form f of x equals m x plus b, where m is the slope and b is the y-intercept. Their graphs are always straight lines. Quadratic functions have the form f of x equals a x squared plus b x plus c, where a cannot be zero. Their graphs are parabolas, which are U-shaped curves. On the right, you can see examples of both: a linear function in blue and a quadratic function in red. Now let's look at the rate of change. For a linear function, the rate of change, or slope, is constant. No matter where you are on the line, the slope remains the same. For our example y equals x plus 1, the slope is always 1. For a quadratic function, the rate of change varies at different points. As we move the point along the parabola y equals x squared minus 1, you can see that the slope of the tangent line changes. The slope equals 2x, which means it depends on the x-coordinate. This varying rate of change is a key difference between linear and quadratic functions. Let's examine the key properties that distinguish linear and quadratic functions. Linear functions always form a straight line and typically have exactly one x-intercept, unless the line is horizontal. They always have exactly one y-intercept and don't have maximum or minimum values. Quadratic functions, on the other hand, always form a parabola or U-shaped curve. They can have up to two x-intercepts, as shown in our example where the parabola crosses the x-axis at negative 1 and positive 1. Quadratics always have exactly one y-intercept and, importantly, they always have exactly one maximum or minimum value, called the vertex. In our example, the minimum occurs at the point (0, -1). Now let's look at some real-world applications and examples of linear and quadratic functions. Linear functions are commonly used to model constant speed motion, where distance is a linear function of time. They're also used in simple interest calculations, fixed-rate pricing models, and many other scenarios where the rate of change is constant. Our example linear function is y equals 2x plus 5, where the slope of 2 represents the constant rate of change, and 5 is the y-intercept. Quadratic functions, on the other hand, are perfect for modeling projectile motion, like the path of a thrown ball, which follows a parabolic trajectory. They're also used in area calculations and optimization problems where we need to find maximum or minimum values. Our example quadratic function is y equals negative 0.5 x squared plus 2x plus 3. Since the coefficient of x squared is negative, this parabola opens downward and has a maximum value at the point (2, 5). To summarize the key differences between linear and quadratic functions: Linear functions, with the form y equals m x plus b, always form straight lines with a constant slope. They're ideal for modeling situations with a constant rate of change. Quadratic functions, with the form y equals a x squared plus b x plus c, always form parabolas with a varying slope at different points. Unlike linear functions which have no maximum or minimum values, quadratic functions always have exactly one extreme value - either a maximum or a minimum. Linear functions are perfect for modeling constant rates of change, such as constant speed motion or fixed-rate pricing. Quadratic functions, on the other hand, are used to model accelerating quantities, such as objects in free fall or optimization problems where we need to find maximum efficiency or minimum cost.