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Problem 38 Easy Difficulty

In Exercises 35-42, use a graphing utility to graph the quadratic function. Identify the vertex, axis of symmetry, and x-intercepts. Then check your results algebraically by writing the quadratic function in standard form.

$ f(x) = x^2 + 10x + 14 $

Answer

Vertex: $(-5,-11)$
Axis of symmetry: $x=-5$
$x$ -intercepts: $(-5 \pm \sqrt{11}, 0)$

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Video Transcript

in this question, we want to use a graphing calculator to find the Vertex axis of symmetry and ex intercepts of given quadratic X squared plus 10 x plus 14. I'm gonna use the desk most graphing calculator to do this. And so if we go out and put this function in X squared plus 10 X plus 14 mhm, the first part of the question is, find the Vertex so the vertex is always the maximum or minimum. And since this quadratic opens up, it is the minimum. Our vertex here is negative. Five. Negative 11 negative five Negative 11. The axis of symmetry is a vertical line of symmetry that crosses the X axis. And, um, it would be is if it's the line that you could fold over in. One half of the Prebble folds onto the other half, and it is always X equals as the vertical line. Thea X coordinate of the Vertex so X equals negative. Five is the axis of symmetry and the ex intercepts all right. About negative 1.683 and negative 8.317 I'm saying about. So it's approximately negative. 8.317 and approximately negative. 1.683 If you need to do this exactly, you can use the quadratic formula or you can complete the square. Um, but the ex intercepts are the solutions to the equation, so we could do negative 10 plus or minus the square root of 10 squared minus four times one times 14 using an a value of wannabe value of Tennessee value of 14 10 squares 100 minus 56 which is 44. And so we're looking at negative 10 plus or minus 44 is four times 11. So would be to square root 11. And, um, I forgot the whole time. Quadratic formula divided by two a. So that would be just dividing by two, which is negative. Five plus reminds the spirit of 11. So we have negative five, plus the square of 11 and negative five miles squared 11. Just in case you needed those to be exact instead of approximations the second, the next part of this question asked us to check the results alphabetically by writing the quadratic function in standard form. And if I think about this with the Vertex of negative five negative 11. That would be X plus five squared minus 11. This tells us that the Vertex would be at negative five negative 11. And if we square X plus five, we get X Square plus 10 X plus 25. You can write it out twice and foil if you need to, or you can square multiply and double square eso X plus five squared is X squared plus 10 X plus 25 that is X squared plus 10 X plus 14, yeah.

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