Answer:
K) I, II, and III
Step-by-step explanation:
Given the quadratic equation in standard form, <em>h </em>= -<em>at</em>² + <em>bt</em> + <em>c</em>, where <em>h </em>is the <u>height</u> or the projectile of a baseball that changes over time, <em>t</em>. In the given quadratic equation, <em>c</em> represents the <u>constant term.</u> Altering the constant term, <em>c</em>, affects the <em>h</em>-intercept, the maximum value of <em>h</em><em>, </em>and the <em>t-</em>intercept of the quadratic equation.
<h2>I. The
<em>h</em>-intercept</h2>
The h-intercept is the value of the height<em>, h</em>, when <em>t = </em>0. This means that setting <em>t</em> = 0 will leave you with the value of the constant term. In other words:
Set <em>t</em> = 0:
<em>h </em>= -<em>at</em>² + <em>bt</em> + <em>c</em>
<em>h </em>= -<em>a</em>(0)² + <em>b</em>(0) + <em>c</em>
<em>h</em> = -a(0) + 0 + <em>c</em>
<em>h</em> = 0 + <em>c</em>
<em>h = c</em>
Therefore, the value of the h-intercept is the value of c.
Hence, altering the value of<em> c </em>will also change the value of the h-intercept.
<h2>II. The maximum value of <em>h</em></h2>
The <u>maximum value</u> of <em>h</em> occurs at the <u>vertex</u>, (<em>t, h </em>). Changing the value of <em>c</em> affects the equation, especially the maximum value of <em>h. </em>To find the value of the <em>t</em>-coordinate of the vertex, use the following formula:
<em>t</em> = -b/2a
The value of the t-coordinate will then be substituted into the equation to find its corresponding <em>h-</em>coordinate. Thus, changing the value of <em>c</em> affects the corresponding <em>h</em>-coordinate of the vertex because you'll have to add the constant term into the rest of the terms within the equation. Therefore, altering the value of <em>c</em> affects the maximum value of <em>h.</em><em> </em>
<h2>III. The <em>t-</em>intercept</h2>
The <u><em>t-</em></u><u>intercept</u> is the point on the graph where it crosses the t-axis, and is also the value of <em>t</em> when <em>h</em> = 0. The t-intercept is the zero or the solution to the given equation. To find the <em>t</em>-intercept, set <em>h</em> = 0, and solve for the value of <em>t</em>. Solving for the value of <em>t</em> includes the addition of the constant term, <em>c</em>, with the rest of the terms in the equation. Therefore, altering the value of <em>c</em> also affects the<em> </em><em>t-intercept</em>.
Therefore, the correct answer is <u>Option K</u>: I, II, and III.
