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y=-(x+6)(x-4) Find the domain and range of the function. Domain All real numbers

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y=-(x+6)(x-4) Find the domain and range of the function. Domain All real numbers

y=-(x+6)(x-4) Find the domain and range of the function. Domain All real numbers

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NortonMaster · Tutor for 5 years

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All real numbers ; y ≤ 27

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## Step 1<br />The given function is a quadratic function, which indicates that y will be any real number depending on x. Thus, since x can indeed be any real number, we can find an x-value that fits into the range.<br /><br />## Step 2<br />For quadratic functions, the shape of the graph is a parabola. If the coefficient of \(x^2\) is positive, the parabola shifts upward; if it’s negative, the parabola shifts downward. Here the coefficient of \(x^2\) (`-1`) is negative, the shape of the quadratic function is an inverted curve, which tells us that it will have a maximum value and then go to negative infinity.<br /><br />## Step 3<br />Hence, for the range, since y can take any value below this maximum, the range will also be the set of all real numbers equal to or below this particular maximum point. The highest point (or y-coordinate of the vertex of the downside parabola) of this function is determined by finding the value of y using the x-coordinate of the vertex of the parabola given by the formula `(-b/2a)`<br /><br />## Step 4<br />So first we find the 'x' which is the axis of symmetry or the x-value of the vertex by using formula -b/(2a). The standard quadratic equation is \(ax^2 + bx + c\) where<br /><br />### **a = -1, b = -2**<br /><br />## Step 5<br />Hence, the 'x' will be `-(-2)/2*(-1) = 1`. <br /><br />## Step 6<br />Putting this x = 1 into the equation will result y = 27 which is the maximum value, as this is an inverted parabola.<br /><br />## Step 7<br />Hence, the range is the set of all real numbers equal to or less than y = `27`.
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