91Ó°ÊÓ

(a) find the vertex and the axis of symmetry of each quadratic function, and determine whether the graph is concave up or concave down. (b) Find the y-intercept and the \(x\) -intercepts, if any. (c) Use parts (a) and (b) to graph the function. (d) Find the domain and the range of the quadratic function. (e) Determine where the quadratic function is increasing and where it is decreasing. (f) Determine where \(f(x)>0\) and where \(f(x)<0\) \(f(x)=-4 x^{2}-6 x+2\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=2 x^{2}+12 x-3\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=4 x^{2}-8 x+3\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=-5 x^{2}+20 x+3\)

Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value, and then find the value. \(f(x)=4 x^{2}-4 x\)

(a) Graph fand g on the same Cartesian plane. (b) Solve \(f(x)=g(x)\) (c) Use the result of part (b) to label the points of intersection of the graphs of fand \(g\). (d) Shade the region for which \(f(x)>g(x)\); that is, the region below fand above \(g\). \(f(x)=-x^{2}+4 ; \quad g(x)=-2 x+1\)

Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -3 and 1 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?

Use the fact that a quadratic function of the form \(f(x)=a x^{2}+b x+c\) with \(b^{2}-4 a c>0\) may also be written in the form \(f(x)=a\left(x-r_{1}\right)\left(x-r_{2}\right),\) where \(r_{1}\) and \(r_{2}\) are the \(x\) -intercepts of the graph of the quadratic function. (a) Find quadratic functions whose \(x\) -intercepts are -5 and 3 with \(a=1 ; a=2 ; a=-2 ; a=5\) (b) How does the value of \(a\) affect the intercepts? (c) How does the value of \(a\) affect the axis of symmetry? (d) How does the value of \(a\) affect the vertex? (e) Compare the \(x\) -coordinate of the vertex with the midpoint of the \(x\) -intercepts. What might you conclude?

A projectile is fired from a cliff 200 feet above the water at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 50 feet per second. The height \(h\) of the projectile above the water is modeled by $$ h(x)=\frac{-32 x^{2}}{50^{2}}+x+200 $$ where \(x\) is the horizontal distance of the projectile from the face of the cliff. (a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the face of the cliff will the projectile strike the water? (d) Graph the function \(h, 0 \leq x \leq 200\). (e) Use a graphing utility to verify the solutions found in parts (b) and (c). (f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?

A projectile is fired at an inclination of \(45^{\circ}\) to the horizontal, with a muzzle velocity of 100 feet per second. The height \(h\) of the projectile is modeled by $$h(x)=\frac{-32 x^{2}}{100^{2}}+x$$ where \(x\) is the horizontal distance of the projectile from the firing point. (a) At what horizontal distance from the firing point is the height of the projectile a maximum? (b) Find the maximum height of the projectile. (c) At what horizontal distance from the firing point will the projectile strike the ground? (d) Graph the function \(h, 0 \leq x \leq 350\). (e) Use a graphing utility to verify the results obtained in parts (b) and (c). (f) When the height of the projectile is 50 feet above the ground, how far has it traveled horizontally?