Question - Quadratic Function Characteristics and Equation
mathquadratic function characteristicsaxis of symmetrygraph of quadratic functionquadratic function equation
Solution:
The question in the image asks you to consider a quadratic function, $$f$$, with the following characteristics:1. Axis of symmetry is the $$y$$-axis,2. The $$x$$-intercepts are $$ (-4, 0) $$ and $$ (4, 0) $$,3. The graph cuts the $$y$$-axis at $$y = 16$$.Followed by two sub-questions:1.1. It asks to sketch the graph of $$f$$ on a system of axes. Clearly show ALL intersepts with the axes.1.2. It asks to determine the equation of the graph in the form $$ax^2 + q = y$$.Let's address each part:For 1.1, sketching the graph requires understanding those characteristics:- The axis of symmetry being the $$y$$-axis implies that the parabola will be mirrored across the $$y$$-axis.- The $$x$$-intercepts are the points where the graph crosses the $$x$$-axis, which are given as $$ (-4, 0) $$ and $$ (4, 0) $$. These are the roots of the quadratic equation.- The graph cuts the $$y$$-axis at $$y = 16$$, which means the $$y$$-intercept is at the point $$ (0, 16) $$.Plotting these points, starting with the $$x$$-intercepts at $$-4$$ and $$4$$, and the $$y$$-intercept at $$16$$, you'll draw a "U"-shaped parabola that opens upwards.For 1.2, to find the equation of the parabola, you can use the vertex form of a quadratic equation, $$y = a(x - h)^2 + k$$, where $$(h, k)$$ is the vertex of the parabola. For this parabola, the vertex is on the $$y$$-axis, so $$h = 0$$. Since the axis of symmetry is the $$y$$-axis, the vertex $$h$$ is 0. Thus, the equation simplifies to $$y = ax^2 + k$$.The $$x$$-intercepts tell us the roots of the quadratic equation are $$x = -4$$ and $$x = 4$$. Therefore, the factored form of the quadratic is \[ y = a(x + 4)(x - 4). \]Expanding this gives us\[ y = a(x^2 - 16). \]We are also told the $$y$$-intercept is $$y = 16$$, so when $$x = 0$$:\[ 16 = a(0 - 16), \]which gives\[ 16 = -16a, \]Hence,\[ a = -1. \]So the equation of the parabola is:\[ y = -1(x^2 - 16), \]which simplifies to\[ y = -x^2 + 16. \]The question asks for a form where $$ax^2 + q = y$$, so this is\[ -x^2 + 16 = y, \]which, when written in the requested form, is\[ y = -x^2 + 16. \]
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