Rewrite the function \(f(x) = 2x^2 - 6x + 5\) in the form \(y = mx + b\).
Answer
However, this is still not in the form \(y = mx + b\). It cannot be expressed in this form because it is a quadratic function, not a linear function.
Step 1 ：First, recognize that \(y = mx + b\) is the slope-intercept form of a linear function. The function \(f(x) = 2x^2 - 6x + 5\) is a quadratic function, not a linear function. In order to rewrite it in slope-intercept form, we will first need to find the vertex of the parabola to complete the square and express it in vertex form \(y = a(x-h)^2 + k\).
Step 2 ：The vertex form of a parabola is \(y = a(x - h)^2 + k\), where \((h, k)\) are the coordinates of the vertex. To find \(h\), we use the formula \(h = -\frac{b}{2a}\). In this case, \(a = 2\) and \(b = -6\), so \(h = -\frac{-6}{2*2} = 1.5\).
Step 3 ：Next, we substitute \(h = 1.5\) into the original equation to find \(k\): \(k = 2*(1.5)^2 - 6*1.5 + 5 = -0.5\).
Step 4 ：Now we can write the function in vertex form: \(f(x) = 2(x - 1.5)^2 - 0.5\).
Step 5 ：However, this is still not in the form \(y = mx + b\). It cannot be expressed in this form because it is a quadratic function, not a linear function.
